Azzouz Dermoune
email: azzouz.dermoune@univ-lille1.fr

Khalifa Es-Sebaiy
email: k.essebaiy@uca.ma

Mohammed Es Sebaiy
email: mohammedsebaiy@gmail.com

Jabrane Moustaaid
email: jabrane.mst@gmail.com

Parametrizations, weights, and optimal prediction: Part 1

Keywords: Parametrization, basis, cubic spline, climate change detection. 1 Motivation

We consider the problem of the annual mean temperature prediction. The years taken into account and the corresponding annual mean temperatures are denoted by 0, . . . , n and t 0 , . . ., t n , respectively. We propose to predict the temperature t n+1 using the data t 0 , . . ., t n . For each 0 ≤ l ≤ n and each parametrization Θ (l) of the Euclidean space R l+1 we construct a list of weights for the data {t 0 , . . . , t l } based on the rows of Θ (l) which are correlated with the constant trend. Using these weights we define a list of predictors of t l+1 from the data t 0 , . . ., t l . We analyse how the parametrization affects the prediction, and provide three optimality criteria for the selection of weights and parametrization. We illustrate our results for the annual mean temperature of France and Morocco.

1

For each 0 ≤ l ≤ n and each parametrization Θ (l) = (θ (l) ji : i, j = 0, . . . , l) of the Euclidean space R l+1 we construct a list of weights for the data {t 0 , . . . , t l } based on the row θ (l) j of Θ (l) which is correlated with the constant trend {1 (l) } ⊤ = (1, . . . , 1), i.e., θ (l) j 1 (l) = 0. We analyze how the parametrization Θ (l) affects the prediction. We also propose a list of criteria for selecting optimal parametrization and weights. We illustrate our results for the annual mean temperature of France and Morocco.

The present paper is the first part of a list of works in preparation. These works are directly related to [START_REF] Dermoune | Estimation of noisy cubic spline using a natural basis[END_REF], [START_REF] Dermoune | Parametrizations, fixed and random effects[END_REF] and [START_REF] Dermoune | General Linear mixed model and signal extraction problem with constraint[END_REF], see also [START_REF] Dermoune | A consistent estimator of the smoothing parameter in the Hodrick-Prescott filter[END_REF], [START_REF] Dermoune | Multivariate Extension of the Hodrick-Prescott Filter-Optimality and Characterization[END_REF], [START_REF] Dermoune | FastICA algorithm: Five criteria for the optimal choice of the nonlinearity function[END_REF].

with i = 0, . . . , l.

If s(i) oscillates around some constant c, then j / ∈I(l) θ

(l)

j s (l) b (l)
ij oscillates around 0. Hence the component j∈I(l) θ (l) j s (l) b (l) ij seems to be the bulk component of s(i), and j / ∈I(l) θ (l)

j s (l) b (l)
ij its residual component. Roughly speaking, the most important coordinates are those correlated with the constant trend 1 (l) , i.e., (θ (l) j : j ∈ I(l)).

3 Conservative rows and selection criteria

Conservative rows

The row p = (p 0 , . . . , p l ) ∈ R l+1 is conservative if

l i=0 p i = 1.
If p i ≥ 0 for all i, then p is a probability distribution on the set {0, . . . , l}. The set of conservative rows is denoted by

M 1 ({0, . . . , l}) = {p ∈ R l+1 : l i=0 p i = 1}.
The mean and the variance of s (l) w.r.t. to p are defined respectively by

m 1 (p) = ps (l) = l i=0 p i s(i), var(p) = l i=0 p i |s(i) -m 1 (p)| 2 .
We have the famous equality

l i=0 p i x 2 i = m 2 1 (p) + var(p).
Observe also as in the probabilistic case, the minimizer

arg min{ l i=0 p i |s(i) -a| 2 : a ∈ R} = m 1 (p),
and the error

min{ l i=0 p i |s(i) -a| 2 : a ∈ R} = var(p).

Selection criterion

The set P contains a finite number of parametrizations of the Euclidean spaces (R 2 , . . . , R n+1 ). An element of P is a parametrization Θ := (Θ (1) , . . . , Θ (n) ) of the Euclidean spaces (R 2 , . . . , R n+1 ).

Let us give for each 1 ≤ l ≤ n and each parametrization Θ (l) a finite subset W (Θ (l) ) of the set of conservative rows M 1 ({0, . . . , l}). We get the subset W (Θ) := W (Θ (1) )×. . .×W (Θ (n) ) of M 1 ({0, 1})×. . .×M 1 ({0, . . . , n}).

A selection criterion S picks a unique element

SW (Θ) = (SW (Θ (1) ), . . . , SW (Θ (n) )) ∈ M 1 ({0, 1}) × . . . × M 1 ({0, . . . , n})
from the set W (Θ).

Prediction cost

We propose for each l = 1, . . . , n, SW (Θ (l) )s (l) as a prediction of s(l + 1). The cost of these predictors for q = 1, 2, +∞, and L < n fixed, is measured by

Cost(SW (Θ), q) = n-1 l=L |s(l + 1) -SW (Θ (l) )s (l) | q n -L , q = 1, 2, := max{|s(l + 1) -SW (Θ (l) )s (l) | : L ≤ l ≤ n -1}, q = +∞.
Let S be a finite set of selection criteria. The optimal selection criterion S * W (Θ * , q) is the minimizer S * W (Θ * , q) = arg min{Cost(SW (Θ), q) : Θ ∈ P, S ∈ S}.

In this work we consider the sets

W (Θ (l) ) = θ (l) j θ (l) j 1 (l) : 0 ≤ j ≤ l, θ (l)
j 1 (l) = 0 , with l = 1, . . . , n. We recall that I(l) = {0 ≤ j ≤ l : θ (l) j 1 (l) = 0}. Observe that for each parametrization Θ (l) , the set I(l) is not empty.

For simplicity we denote for each selection criterion S S(Θ (l) ) := SW (Θ (l) ), l = 1, . . . , n.

Now, we are going to define our selection criteria.

The selection criterion S u

Let I(l) = {j(0), . . . , j(card(I(l)) -1)} be the elements of the set I(l) with j(0) < . . . < j(card(I(l)) -1). We define for each u ≤ n fixed the selection criterion

S u (Θ (l) ) = θ (l) j(min(u,card(I(l))-1)) θ (l)
j(min(u,card(I(l))-1)) 1 (l) .

If u = 0, then S 0 (Θ (l) ) = θ (l) j(0) θ (l) j(0) 1 (l)
.

If u = n, then S n (Θ (l) ) = θ (l) j(card(I(l))-1) θ (l) j(card(I(l))-1) 1 (l)
.

3.5

The winning conservative rows S u(q,Θ) (Θ)

Given Θ and q = 1, 2, +∞, the optimal selection criterion among (S u : u = 0, . . . , n) is given by the minimizer u(q, Θ) = arg min{Cost(S u (Θ), q) : u = 0, . . . , n}.

Hence S u(q,Θ) (Θ) is the optimal conservative rows among the set (S u (Θ) : u = 0, . . . , n) of conservative rows.

The selection criterion S mean

For 1 ≤ l ≤ n, we consider the selection criterion

S mean (Θ (l) ) = j∈I(l) θ (l) j card(I(l))θ (l) j 1 (l)
.

Observe that for the canonical parametrization Θ (l) = (δ j i : 0 ≤ i, j ≤ l), we have

S mean (Θ (l) ) = ( 1 l + 1 , . . . , 1 l + 1
).

3.7

The winning conservative rows S 1,q (Θ)

Given Θ and q = 1, 2, +∞, the optimal conservative rows among S u(q,Θ) (Θ) and S mean (Θ) is the minimizer S 1,q (Θ) := arg min{Cost(S u(q,Θ) (Θ), q), Cost(S mean (Θ), q)}.

The selection criterion S utail1

For a fixed u ≤ n the set

J 1 (l, u) = arg max{ l i=min(u,l) θ (l) ji θ (l) j 1 (l) : j ∈ I(l)}
may be not a singleton. It furnishes the selection criterion

S tail1 (Θ (l) ) = j∈J 1 (l,u) θ (l) j card(J 1 (l, u))θ (l) j 1 (l) .
As a simple example, if Θ (l) is the canonical parametrization and u = n, then

J 1 (l, n) = l. If u = n -1, then J 1 (l, n -1) = l for l ≤ n -1, and J 1 (n, n -1) = {n -1, n}.
3.9 The winning conservative rows S u(q,Θ)tail1 (Θ)

The optimal selection criterion among (S utail1 : u = 0, . . . , n) is the minimizer of S u(q,Θ)tail1 = arg min{Cost(Θ, S utail1 (Θ), q) : u = 0, . . . , n}.

3.10

The winning conservative rows S 2,q (Θ)

The winner for each q = 1, 2, +∞ fixed, among S u(q,Θ)tail1 (Θ) and S 1,q (Θ) is the minimizer S 2,q (Θ) := arg min{Cost(S u(q,Θ)tail1 (Θ), q), Cost(S 1,q (Θ), q)}.

The selection criterion S utail2

For a fixed u ≤ n the set

J 2 (l, u) = arg max{max{ θ (l) ji θ (l) j 1 (l) : min(u, l) ≤ i ≤ l} : j ∈ I(l)},
may be not a singleton. It furnishes the selection criterion

S tail2 (Θ (l) ) = j∈J 2 (l,u) θ (l) j card(J 2 (l, u))θ (l) j 1 (l)
.

3.12 The winning conservative rows S u(q,Θ)tail2 (Θ)

The optimal selection criterion among (S utail2 : u = 0, . . . , n) is the minimizer of S u(q,Θ)tail2 = arg min{Cost(S utail2 (Θ), q) : u = 0, . . . , n}.

3.13

The winning conservative rows S 3,q (Θ)

The winner for each q = 1, 2, +∞ fixed, among S u(q,Θ)tail2 (Θ) and S 2,q (Θ) is the minimizer S 3,q (Θ) := arg min{Cost(S u(q,Θ)tail2 (Θ), q), Cost(S 2,q (Θ), q)}.

The selection criterion S maxcor

The set

J 3 (l) = arg max{ |θ (l) j 1 (l) | √ l + 1 θ (l) j : j ∈ I(l)}
of the coordinates θ (l) j highly correlated with the constant trend 1 (l) , furnishes the selection criterion

S maxcor (Θ (l) ) = j∈J 3 (l) θ (l) j card(J 3 (l))θ (l) j 1 (l) .
3.15 The winning conservative rows S 4,q (Θ)

The winner for each q = 1, 2, +∞ fixed, among S maxcor (Θ) and S 3,q (Θ) is the minimizer S 4,q (Θ) := arg min{Cost(S maxcor (Θ), q), Cost(S 3,q (Θ), q)}.

The selection criterion S qnearU

We consider the set

J 4 (l, q) = arg min{ θ (l) j θ (1) j 1 (l) -( 1 l + 1 , . . . , 1 l + 1 ) q : j ∈ I(l)} of the nearest conservative rows θ (l) j θ (1)
j 1 (l) to the uniform conservative row ( 1 l+1 , . . . , 1 l+1 ), and the corresponding selection criterion

S qnearU (Θ (l) ) = j∈J 4 (l,q) θ (l) j card(J 4 (l, q))θ (l) j 1 (l)
.

Here • q denotes the l(q)-norm with q = 1, 2, +∞.

3.17 The winning conservative rows S q(Θ)nearU (Θ)

For each q = 1, 2, +∞ fixed let us denote by q(Θ) = arg min{Cost(S q 1 nearU (Θ), q) : q 1 = 1, 2, +∞}, and then we obtain the winning conservative rows S q(Θ)nearU (Θ) among the three conservative rows (S q 1 nearU (Θ) : q 1 = 1, 2, +∞).

3.18

The winning conservative rows S 5,q (Θ)

For each q = 1, 2, +∞ fixed, the winner among S q(Θ)nearU (Θ) and S 4,q (Θ) is the minimizer S 5,q (Θ) := arg min{Cost(S q(Θ)nearU (Θ), q), Cost(S 4,q (Θ), q)}.

The selection criterion S uvar

For each j ∈ I(l) the variance of the data s (l) w.r.t. to the conservative row θ (l) j θ (l) j 1 (l) is denoted by var(l, j). We define the one-to-one map σ (l) from {0, . . . , card(I(l)) -1} to I(l) as follows. The integer σ (l) (0) is the first element of arg min{var(l, j) : j ∈ I(l)}.

By induction for k < card(I(l)) -1 the integer σ (l) (k + 1) is the first element of arg min{var(l, j) : j ∈ I(l) \ {σ (l) (0), . . . , σ (l) (k)}}.

We define for a fixed u ≤ n the index j(l, u) = σ (l) (min(u, card(I(l)) -1)), and the selection criterion

S uvar (Θ (l) ) = θ (l) j(l,u) θ (l) j(l,u) 1 (l)
.

If u = n, then j(l, n) = σ (l) (card(I(l))-1) is the index of the largest variance. If u = 0, then j(l, 0) = σ (l) (0) is the index of the smallest variance.

3.20

The winning conservative rows S u(q,Θ)var (Θ)

The optimal selection criterion among (S uvar : u = 0, . . . , n) is the minimizer of S u(q,Θ)var = arg min{Cost(Θ, S uvar (Θ), q) : u = 0, . . . , n}.

3.21

The winning conservative sequence S 6,q (Θ)

The winner for each q = 1, 2, +∞ fixed, among S u(q,Θ)var (Θ) and S 5,q (Θ) is the minimizer

3.23

The winning conservative rows S u(q,Θ)v(q,Θ)f d (Θ)

Given Θ and q = 1, 2, +∞, the optimal conservatrice sequence S u(q,Θ)v(q,Θ)f d (Θ) among (S uvf d (Θ) : u, v = 0, . . . , n) is given by the minimizer (u(q, Θ), v(q, Θ)) = arg min{Cost(S uvf d (Θ), q) : u, v = 0, . . . , n}.

3.24

The winning conservative rows S 7,q (Θ)

Given Θ and q = 1, 2, +∞, the optimal conservative rows S 7,q (Θ) among S u(q,Θ)v(q,Θ)fd (Θ) and S 6,q (Θ) is the minimizer arg min{Cost(S u(q,Θ)v(q,Θ)fd (Θ), q, L), Cost(S 6,q (Θ), q)}.

4 The winning conservative rows S 7 (Θ(q))

We constructed for q = 1, 2, +∞ fixed and each parametrization Θ the optimal conservative rows S 7,q (Θ). Assume that we have a finite set P of parametrizations. The minimizer Θ * (q) of the map Θ → Cost(S 7,q (Θ), q)

furnishes the optimal selection criterion S 7 (Θ(q)) := S 7,q (Θ(q)).

Application to parametrizations given by the energy of the spline

We identify for the integer l ≥ 1 the space R l+1 with the space of the natural cubic splines S 3,nat (0, . . . , l) having the knots 0, . . . , l. Let us denote S 3 (0, . . . , l) the set of cubic splines having the knots 0, . . . , l. We recall that an element s ∈ S 3 is a C 2 map on [0, l] and is a polynomial of degree three on each interval [i, i + 1) for i = 0,. . . , l -1. More precisely, let

p 0 = s(0), . . . , p l = s(l), q 0 = s ′ (0), . . . , q l = s ′ (l), u 0 = s ′′ (0), . . . , u l = s ′′ (l), v 0 = s ′′′ (0+), . . . , v l-1 = s ′′′ (l -1+)
be respectively the values of s and its derivatives up to order three on the knots. We have for i = 0, . . . , l -1,

s(t) = p i + q i (t -i) + (t -i) 2 u i /2 + (t -i) 3 v i /6, t ∈ [i, i + 1).
The following constraint guarantees the hypothesis that s is C 2 :

p i + q i + u i /2 + v i /6 = p i+1 , (5.1)
q i + u i + v i /2 = q i+1 , (5.2)
v i = s (3) (t i ) = u i+1 -u i .
(

It is well known [START_REF] Deboor | A Practical Guide to Spline[END_REF] that S 3 (0, . . . , l) has the dimension l + 3, see also [START_REF] Craven | Smoothing noisy data with spline functions[END_REF] and [START_REF] Wahba | Spline models for observational data[END_REF]. Hence an element s ∈ S 3 (0, . . . , l) is completely defined by l + 3 independent parameters. Moreover, the set of natural cubic splines S 3,nat (0, . . . , l) is the set of cubic spline s with s ′′ (0) = s ′′ (l) = 0. Hence the dimension of S 3,nat (0, . . . , l) is equal to l + 1. Now we are ready to define our parametrizations of R l+1 . There exist for each l fixed a unique non symmetric matrix M (l) and a unique symmetric matrix S (l) such that

l 0 |s(t)| 2 dt = {s (l) } ⊤ M (l) s (l) = {s (l) } ⊤ S (l) s (l) ,
for all s ∈ S 3,nat (0, . . . , l).

We consider the following six parametrization matrices

Θ (l) = M (l) , {M (l) } ⊤ , {M (l) } -1 , {{M (l) } -1 } ⊤ , S (l) , {S (l) } -1 .

Real data application

In the temperature prediction problem we are interested in the annual mean temperature observed in France and Morocco from 1901 to 2015. Data s (n) = (s(0), . . . , s(n)) ⊤ with n = 114 respectively for France and Morocco are presented in Figure (3). Observe that s(n) denotes the temperature of the year n + 1.

Predictors

Our set of parametrizations P contains

M = (M (l) : l = 1, . . . , n), M ⊤ = ({M (l) } ⊤ : l = 1, . . . , n), M -1 = ({M (l) } -1 : l = 1, . . . , n), {M -1 } ⊤ = ({{M (l) } -1 } ⊤ : l = 1, . . . , n), S = (S (l) : l = 1, . . . , n), S -1 = ({S (l) } -1 : l = 1, . . . , n).
Table [START_REF] Craven | Smoothing noisy data with spline functions[END_REF] shows that for each q = 1, 2, +∞ and the lag L = 4 the optimal parametrization Θ(q) = M -1 for both France and Morocco, but the optimal conservative rows S 7 (Θ(q)) do not coincide. The optimal conservative rows S 7 (Θ (114) (q)) are plotted in Figures [START_REF] Dermoune | Multivariate Extension of the Hodrick-Prescott Filter-Optimality and Characterization[END_REF] and [START_REF] Dermoune | Estimation of noisy cubic spline using a natural basis[END_REF]. The predictors of the temperature s(114) (the temperature at the year 2015) and the true temperature is given in Table [START_REF] Deboor | A Practical Guide to Spline[END_REF]. The predictors of the temperature s(115) (the temperature at the year 2016) is given in Table [START_REF] Dermoune | A consistent estimator of the smoothing parameter in the Hodrick-Prescott filter[END_REF]. Splines of the true temperature and its optimal predictors are represented in Figure [START_REF] Dermoune | Parametrizations, fixed and random effects[END_REF]. M -1 M -1 M -1 S 7 (Θ(q)) S utail1 (u = 73) S utail2 (u = 41) cost 0.6183125 0.6027288 1.917094

Country France q 1 2 ∞ Θ(q) M -1 M -1 M -1
Table 1: The optimal choice Θ(q) and S 7 (Θ(q)). The best prediction of s(114) using the optimal parametrization Θ(q) = M -1 and the optimal conservative rows S 7 (Θ(q)). n) and its optimal predictors, with n = 114.

Country

Conclusion.

Having a time series s(0), . . ., s(n) with values in R, we showed how to predict the value s(n + 1) from each parametrization of the set R n+1 . We also provided optimality criteria to select the best predictor. This work can be extended to time series s(i) ∈ K with K is any field or vector space.

Figure 1 :Figure 2 :

 12 Figure 1: Represantation of the basis B (l) = M (l) with l = 7.

Figure 3 :

 3 Figure 3: Annual mean temperatures in France and Morocco from 1901 to 2015

  S 7 (Θ(q)) S utail2 (u = 86) S uvf d (u = 93, v = 5) S uvf d (u = 81, v = 8)

Figure 6 :

 6 Figure6: The splines of s(n) and its optimal predictors, with n = 114.

Table 2 :

 2

The best prediction of s(115) (the temperature at the year 2016) using the optimal parametrization Θ(q) = M -1 and the optimal conservative rows S 7 (Θ(q)). The optimal conservative row S 7 (Θ (114) (q)), q = 1, 2, ∞ for France.


Brian Cook

Kevin Hughes

Zane Kun Li
email: zkli@ncsu.edu

Akshat Mudgal

Olivier Robert

Po-Lam Yung

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory. M S C 2 0 2 0 42B15, 43A25, 43A70 (primary), 11L07 (secondary)

INTRODUCTION

Motivation

Let 𝑠 ⩾ 1 and 𝑘 ⩾ 2 be integers. For 𝑋 ⩾ 1, let 𝐽 𝑠,𝑘 (𝑋) be the number of solutions to the degree 𝑘 Vinogradov system in 2𝑠 variables:

𝑥 𝑗 1 + 𝑥 𝑗 2 + ⋯ + 𝑥 𝑗 𝑠 = 𝑦 𝑗 1 + 𝑦 𝑗 2 + ⋯ + 𝑦 𝑗 𝑠 , 1 ⩽ 𝑗 ⩽ 𝑘, (1.1)
where all variables 𝑥 1 , … , 𝑥 𝑠 , 𝑦 1 , … , 𝑦 𝑠 ∈ [1, 𝑋] ∩ ℕ. Nontrivial upper bounds for 𝐽 𝑠,𝑘 (𝑋) were first studied by Vinogradov in 1935 [START_REF] Vinogradov | New estimates for Weyl sums[END_REF] and such results are collectively referred to as Vinogradov's mean value theorem (VMVT) in the literature. The main conjecture in VMVT, now a theorem as of 2015, was that for every 𝜀 > 0 and 𝑠, 𝑘 ∈ ℕ, one has 𝐽 𝑠,𝑘 (𝑋) ≲ 𝑠,𝑘,𝜀 𝑋 𝜀 ( 𝑋 𝑠 + 𝑋 2𝑠-𝑘(𝑘+1)

2

) (1.2)
for all 𝑋 ⩾ 1. It is not hard to see that 𝐽 𝑠,𝑘 (𝑋) ≳ 𝑠,𝑘 𝑋 𝑠 + 𝑋 2𝑠-𝑘(𝑘+1)∕2 and applying Hölder's inequality, we may deduce (1.2) for all 𝑠 ∈ ℕ from the 𝑠 = 𝑘(𝑘 + 1)∕2 case. VMVT plays an important role in understanding Waring's problem and the Riemann zeta function, see, for example, [START_REF] Ford | Vinogradov's integral and bounds for the Riemann zeta function[END_REF][START_REF] Ford | Zero-free regions for the Riemann zeta function, Number theory for the millennium[END_REF][START_REF] Heath-Brown | A new 𝑘th derivative estimate for exponential sums via Vinogradov's mean value[END_REF][START_REF] Wooley | Translation invariance, exponential sums, and Waring's problem[END_REF]. When 𝑘 = 2, the main conjecture in VMVT is classical. In 2014, Wooley [START_REF] Wooley | The cubic case of the main conjecture in Vinogradov's mean value theorem[END_REF] proved the 𝑘 = 3 case of VMVT using the method of efficient congruencing (see also [START_REF] Heath-Brown | The cubic case of Vinogradov's mean value theorem: a simplified approach to Wooley's "efficient congruencing[END_REF] for a shorter proof due to Heath-Brown). In 2015, the 𝑘 ⩾ 2 case was proven by Bourgain, Demeter, and Guth in [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF] using Fourier decoupling for the degree 𝑘 moment curve from which VMVT followed as a corollary. Finally, in 2017, Wooley [START_REF] Wooley | Nested efficient congruencing and relatives of Vinogradov's mean value theorem[END_REF], gave an alternative proof of (1.2) for all 𝑘 ⩾ 2 using nested efficient congruencing.

After the proofs of VMVT using the Fourier method of decoupling [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF] and the number theoretic method of efficient congruencing [START_REF] Wooley | Nested efficient congruencing and relatives of Vinogradov's mean value theorem[END_REF], it has been an interesting question to determine how these two methods are related and whether a "dictionary" between the two methods could be obtained. The study of this dictionary has led to new proofs of Fourier decoupling for the parabola [START_REF] Li | An 𝑙 2 decoupling interpretation of efficient congruencing: the parabola[END_REF], cubic moment curve [START_REF] Guo | A bilinear proof of decoupling for the cubic moment curve[END_REF], and the degree 𝑘 moment curve [START_REF] Guo | A short proof of 𝓁 2 decoupling for the moment curve[END_REF]; these having been inspired from the efficient congruencing arguments in [26, section 4], [START_REF] Heath-Brown | The cubic case of Vinogradov's mean value theorem: a simplified approach to Wooley's "efficient congruencing[END_REF], and [START_REF] Wooley | Nested efficient congruencing and relatives of Vinogradov's mean value theorem[END_REF], respectively. Additionally, a decoupling interpretation of the study of VMVT over ellipspephic sets [START_REF] Biggs | Efficient congruencing in ellipsephic sets: the quadratic case[END_REF] led to a proof of Fourier decoupling for fractal sets on the parabola [START_REF] Chang | Decoupling for fractal subsets of the parabola[END_REF].

In this article, we revisit a particular classical VMVT which states that 𝐽 𝑠,𝑘 (𝑋) ≲ 𝑠,𝑘 𝑋 2𝑠-𝑘(𝑘+1) 2 + 1 2 𝑘 2

( 1-1

𝑘

) 𝑠∕𝑘 (1.3) for all 𝑋 ⩾ 1 and 𝑠 = 𝑘𝑙 with 𝑙 ∈ ℕ. This result should be compared to the supercritical 𝑠 ⩾ 𝑘(𝑘 + 1)∕2 case in (1.2). For 𝑠 very large compared to 𝑘, we have an extra term 1 2 𝑘 2 (1 -1 𝑘 ) 𝑠∕𝑘 in the exponent, which decays exponentially in 𝑠 for every fixed value of 𝑘, instead of an 𝜀. The estimate (1.3) appears (for example) in Vaughan's book [31, chapter 5] and is a refinement of an argument of Karatsuba [START_REF] Karatsuba | Mean value of the modulus of a trigonometric sum[END_REF] from 1973 (see also Stechkin [START_REF] Stečkin | Mean values of the modulus of a trigonometric sum[END_REF] from 1975). The loss of the 𝑋 The main purpose of this paper is to illustrate how this refined argument of Karatsuba can be adapted to give a proof of a nonsharp Fourier decoupling inequality for the degree 𝑘 moment curve in the supercritical regime. The key difficulty that prevents the direct use of ideas from [START_REF] Guo | A bilinear proof of decoupling for the cubic moment curve[END_REF][START_REF] Guo | A short proof of 𝓁 2 decoupling for the moment curve[END_REF][START_REF] Li | An 𝑙 2 decoupling interpretation of efficient congruencing: the parabola[END_REF] is the heavy reliance on solution counting in (1.3). One of the main points of this article is to clarify the role of such solution counting arguments in the study of Fourier decoupling. The mechanism driving the solution counting arguments will allow us to prove the key Lemma 4.4, which concerns the geometry of Fourier supports of the functions appearing in our main Theorem 1.1.

As our goal is to clarify the role of solution counting in Fourier decoupling and Bourgain, Demeter, and Guth have already given the sharpest possible moment curve decoupling theorem in [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF], we will work over ℚ 𝑞 rather than over ℝ. This will allow us to present the argument in the cleanest possible manner, free of technical difficulties arising from the inconvenience of the uncertainty principle in ℝ 𝑘 . See also [START_REF] Guo | Improved discrete restriction for the parabola[END_REF] for another decoupling paper that works over ℚ 𝑞 rather than ℝ, there however, the authors use the observation that decoupling over ℚ 𝑞 is quantitatively more efficient than decoupling over ℝ in terms of exponential sum estimates.

Notation

As 𝑘 will be fixed, we will allow all constants to depend on 𝑘. Given two positive expressions 𝑋 and 𝑌, we write 𝑋 ≲ 𝑌 if 𝑋 ⩽ 𝐶𝑌 for some constant 𝐶 that is allowed to depend on 𝑘. If 𝐶 depends on some additional parameter 𝐴, then we write 𝑋 ≲ 𝐴 𝑌. We write 𝑋 ∼ 𝑌 if 𝑋 ≲ 𝑌 and 𝑌 ≲ 𝑋. By writing 𝑓(𝑥) = 𝑂(g(𝑥)), we mean |𝑓(𝑥)| ≲ g(𝑥). We say that 𝑓 has Fourier support in a set Ω if its Fourier transform f is supported in Ω.

To prepare the reader for the myriad of intervals that will occur later in Sections 4 and 5, there will be three types of interval lengths: intervals named with a "𝐾" will be associated to the smallest scale 𝛿, intervals named with a "𝐽" will be associated to the intermediate scale 𝜈 ≈ 𝛿 1∕𝑘 , and intervals named with an "𝐼" will be associated to the largest scale 𝜅 ≈ 𝛿 𝜀 (though on a first reading, it might be easier to set 𝜅 = 1∕𝑞). Finally, in the context of the decoupling constant 𝔇 𝑝 (𝛿), defined in (1.5), we call 𝑝 subcritical if 𝑝 < 𝑘(𝑘 + 1) and 𝑝 supercritical if 𝑝 ⩾ 𝑘(𝑘 + 1) (rather than the more accurate but slightly more clumsy "not subcritical").

Analysis over ℚ 𝒒 and decoupling

Fix a degree 𝑘 ⩾ 2 and a prime number 𝑞 with 𝑞 > 𝑘. We reserve the letter 𝑝 for the Lebesgue exponent in the main Theorem 1.1. We very briefly review the harmonic analysis over ℚ 𝑞 needed to set up the statement of decoupling. See also Section 2 and [14, section 2] for further discussion surrounding the harmonic analysis and basic geometric facts over ℚ 𝑞 that are useful in decoupling. Additionally see chapters 1 and 2 of [START_REF] Taibleson | Fourier analysis on local fields[END_REF] and chapter 1 (in particular sections 1 and 4) of [START_REF] Vladimirov | 𝑝-adic analysis and mathematical physics[END_REF] for a more complete discussion of analysis on ℚ 𝑞 . The field ℚ 𝑞 is the completion of ℚ under the 𝑞-adic norm, defined by |0| = 0 and |𝑞 𝑎 𝑏∕𝑐| = 𝑞 -𝑎 if 𝑎 ∈ ℤ, 𝑏, 𝑐 ∈ ℤ ⧵ {0} and 𝑞 is relatively prime to both 𝑏 and 𝑐. Then ℚ 𝑞 can be identified (bijectively) with the set of all formal series is then a set of the form {𝜉 ∈ ℚ 𝑞 ∶ |𝜉 -𝑎| ⩽ 𝑟}, where 𝑎 ∈ ℚ 𝑞 and 𝑟 ⩾ 0; 𝑟 will then be called the length of the interval. We also will use |𝐼| to denote the length of an interval 𝐼. The ring of integers ℤ 𝑞 coincides with the unit interval {𝜉 ∈ ℚ 𝑞 ∶ |𝜉| ⩽ 1}. A cube in ℚ 𝑘 𝑞 of side length 𝑟 is then a product of 𝑘 intervals in ℚ 𝑞 of lengths 𝑟. We will work with Schwartz functions defined on ℚ 𝑘 𝑞 (i.e., finite linear combinations of characteristic functions of cubes in ℚ 𝑘 𝑞 ). The Fourier transform of such a function 𝑓 will be given by

ℚ 𝑞 = { ∞ ∑ 𝑗=𝑘 𝑎 𝑗 𝑞 𝑗 ∶ 𝑘 ∈ ℤ, 𝑎 𝑗 ∈ {0,
f(𝜉) ∶= ∫ ℚ 𝑘 𝑞 𝑓(𝑥)𝜒(-𝑥 ⋅ 𝜉)𝑑𝑥,
where 𝜒 is a fixed element in the Pontryagin dual Q𝑞 of ℚ 𝑞 that restricts to the principal character on the additive subgroup ℤ 𝑞 and restricts to a nonprincipal character on the additive subgroup For 𝛿 ∈ 𝑞 -ℕ and any interval 𝐼 ⊂ ℚ 𝑞 with length ⩾ 𝛿, let 𝑃 𝛿 (𝐼) be a partition of 𝐼 into intervals of length 𝛿. Write 𝑃 𝛿 for 𝑃 𝛿 (ℤ 𝑞 ). To each interval 𝐼 ⊂ ℤ 𝑞 , one associates a parallelepiped

𝑞 -1 ℤ 𝑞 , 𝑥 ⋅ 𝜉 = ∑ 𝑘 𝑖=1 𝑥 𝑖 𝜉 𝑖 if 𝑥 = (𝑥
𝜃 𝐼 ∶= { 𝛾(𝑎) + 𝑘 ∑ 𝑗=1 𝑡 𝑗 𝛾 (𝑗) (𝑎) ∈ ℚ 𝑘 𝑞 ∶ |𝑡 𝑗 | ⩽ |𝐼| 𝑗 for all 1 ⩽ 𝑗 ⩽ 𝑘 } of dimensions |𝐼| × |𝐼| 2 × ⋯ × |𝐼| 𝑘
, where 𝑎 ∈ 𝐼; this parallelepiped is independent of the choice of 𝑎 ∈ 𝐼. Note that ⋃ 𝐾∈𝑃 𝛿 𝜃 𝐾 is a covering of a 𝛿 𝑘 neighborhood of the unit moment curve (in fact it covers a suitable anisotropic neighborhood of that curve). One also associates to each 𝐾 ∈ 𝑃 𝛿 a cube

𝜏 𝐾 ∶= {(𝜉 1 , … , 𝜉 𝑘 ) ∈ ℚ 𝑘 𝑞 ∶ |𝜉 𝑗 -𝑎 𝑗 | ⩽ 𝛿 for all 1 ⩽ 𝑗 ⩽ 𝑘} (1.4)
of side length 𝛿, where 𝑎 ∈ 𝐾; again this is independent of the choice of 𝑎 ∈ 𝐾. Note that for each 𝐾 ⊂ 𝑃 𝛿 , the ultrametric inequality gives that 𝜃 𝐾 ⊂ 𝜏 𝐾 .

For an interval 𝐼 ⊂ ℤ 𝑞 , let 𝑓 𝐼 be defined such that f𝐼 ∶= f ⋅ 1 𝐼×ℚ 𝑘-1

𝑞

. For 𝑝 ⩾ 2 and 𝛿 ∈ 𝑞 -ℕ , let 𝔇 𝑝 (𝛿) be the smallest constant such that the inequality

‖𝑓‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽ 𝔇 𝑝 (𝛿) ( ∑ 𝐾∈𝑃 𝛿 ‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )
) 1∕2

(1.5) [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF] showed that

𝔇 𝑝 (𝛿) ≲ 𝜀,𝑝,𝑞 𝛿 -𝜀 ( 1 + 𝛿 -( 1 2 -𝑘(𝑘+1) 2𝑝 ) ) , (1.6)
and this estimate is sharp. Strictly speaking [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF] proves a decoupling theorem over ℝ rather than over ℚ 𝑞 , but the same proof can be used to derive (1.6). Choosing 𝑓 to be a sum of Dirac deltas immediately implies (1.2).

The main result

By interpreting the refinement of Karatsuba's argument for (1.3) into decoupling language, our main result is then the following Fourier decoupling analogue of (1.3). In the same way that (1.3) is a weaker partial result toward (1.2), Theorem 1.1 and Corollary 1.2 should be viewed as the analogous weaker counterpart of the sharp bound (1.6).

Theorem 1.1. Let 𝑝 0 ∈ 2ℕ be an even integer and let 𝑐(𝑝 0 ) ⩾ 0 be such that

𝔇 𝑝 0 (𝛿) ⩽ 𝐶 1 𝛿 -( 1 2 -𝑘(𝑘+1) 2𝑝 0 )- 𝑐(𝑝 0 ) 𝑝 0 (1-1 𝑘 ) 𝑝 0 ∕(2𝑘) for all 𝛿 ∈ 𝑞 -ℕ (1.7)
where 𝐶 1 is independent of 𝛿. If 𝑝 ∈ 𝑝 0 + 2𝑘ℕ and 0 < 𝜀 < 1, then

𝔇 𝑝 (𝛿) ≲ 𝑝,𝜀,𝐶 1 𝑞 𝑎(𝑝,𝑝 0 )∕𝑝 𝛿 -( 1 2 -𝑘(𝑘+1) 2𝑝 )- 𝑐(𝑝 0 ) 𝑝 (1-1 𝑘 ) 𝑝∕(2𝑘) -𝜀 for all 𝛿 ∈ 𝑞 -ℕ , (1.8)
where

𝑎(𝑝, 𝑝 0 ) ∶= ( 𝑝 -𝑝 0 2𝑘 ) ( 𝑝 0 2 + 𝑘 2 + 7𝑘 -4 2
) + 𝑘 2 ( 𝑝 -𝑝 0 2𝑘 )( 𝑝 -𝑝 0 2𝑘 + 1
) .

(1.9)

As 𝔇 𝑝 (𝛿) ⩾ 1 for all 𝑝, (1.7) implies that 𝑐(𝑝 0 ), 𝑘, and 𝑝 0 are such that

1 2 - 𝑘(𝑘 + 1) 2𝑝 0 + 𝑐(𝑝 0 ) 𝑝 0 ( 1 - 1 𝑘 ) 𝑝 0 2𝑘 ⩾ 0. (1.10)
It is also known that 𝔇 2𝑘 (𝛿) ≲ 𝜀 𝛿 -𝜀 for any 𝜀 > 0, see, for example, [START_REF] Demeter | Fourier restriction, decoupling, and applications[END_REF]Exercise 11.19] for the Euclidean case; we provide a proof for the case over ℚ 𝑞 in the Appendix for the convenience of the reader. We also remark that [START_REF] Hickman | A non-archimedean variant of Littlewood-Paley theory for curves[END_REF] proved, in the case of local fields, a related square function estimate with a bound independent of 𝛿 if the 𝑓 𝐾 's are Fourier supported in a 𝛿 𝑘 neighborhood of 𝛾(𝐾); see also [START_REF] Gressman | Reversing a philosophy: from counting to square functions and decoupling[END_REF] and [START_REF] Biggs | Reinforcing a philosophy: a counting approach to square functions over local fields[END_REF] for similar estimates. Choosing 𝑝 0 = 2𝑘 and 𝑐(𝑝 0 ) = 𝑘 2 ∕2 + 𝜀 for any 𝜀 > 0 in applying Theorem 1.1 we obtain:

Corollary 1.2. Let 𝑝 ∈ 2𝑘ℕ and 0 < 𝜀 < 1. Then 𝔇 𝑝 (𝛿) ≲ 𝑝,𝜀 𝑞 𝑂(𝑘+𝑝∕𝑘) 𝛿 - ( 1 2 -𝑘(𝑘+1) 2𝑝 ) -𝑘 2 2𝑝 (1-1 𝑘 ) 𝑝∕(2𝑘) -𝜀 for all 𝛿 ∈ 𝑞 -ℕ ,
where the implied constant in the exponent of 𝑞 is absolute (and independent of 𝑘). The exponent of 𝑞 in Corollary 1.2 is more precisely

𝑎(𝑝,2𝑘) 𝑝 = ( 1 2𝑘 -1 𝑝 ) 𝑘 2 +9𝑘-4 2 + 1 4 ( 𝑝 2𝑘 -1
), but we opt to write it as above because it more clearly illustrates what the main terms are. Note that the hypothesis in Theorem 1.1 is always satisfied if 𝑝 0 is any fixed exponent ⩾ 2 and 𝑐(𝑝 0 ) is chosen large enough. One can view Theorem 1.1 as a way of upgrading trivial 𝑙 2 𝐿 𝑝 0 decoupling at say some subcritical 𝑝 to 𝑙 2 𝐿 𝑝 decoupling for all large 𝑝 with only a loss that decreases exponentially as 𝑝 → +∞. Of course, if one already knew the sharp estimate in the critical 𝑝 0 = 𝑘(𝑘 + 1) case, then Theorem 1.1 implies that we know the sharp decoupling estimate for all 𝑝 ∈ 𝑘(𝑘 + 1) + 2𝑘ℕ. However, this already follows from interpolating the critical estimate with the trivial 𝑙 2 𝐿 ∞ decoupling estimate.

Though Corollary 1.2 implies (1.3) with an extra 𝑋 𝜀 that comes from the 𝛿 -𝜀 factor in Corollary 1.2, Corollary 1.2 is more general and this extra 𝛿 -𝜀 term comes from needing some additional uniformity in the case of the general 𝑓 Fourier supported in ⋃ 𝐾∈𝑃 𝛿 𝜃 𝐾 and an application of the broad-narrow argument to get around the use of the Prime Number Theorem in the proof of (1.3) (see Subsection 4.1.1). See Subsections 3.5 and 5.1 for some more discussion comparing the VMVT case and the general 𝑓 decoupling case.

We end with some discussion about how the proof of Corollary 1.2 (and Theorem 1.1) contrasts with modern decoupling proofs of degree 𝑘 moment curve decoupling [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF][START_REF] Guo | A short proof of 𝓁 2 decoupling for the moment curve[END_REF] that prove (1.6). Unlike the argument in [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF][START_REF] Guo | A short proof of 𝓁 2 decoupling for the moment curve[END_REF], we are missing any lower dimensional decoupling input and while we do use induction on scales, the iteration itself is unique in that it iterates on the 𝑝 in 𝑙 2 𝐿 𝑝 decoupling. Schematically, the iteration to prove Theorem 1.1 controls 𝑙 2 𝐿 𝑝 decoupling by 𝑙 2 𝐿 𝑝-2𝑘 decoupling at a larger scale. After 𝑂(𝑝∕𝑘) steps, we are reduced to 𝑙 2 𝐿 2𝑘 decoupling for the degree 𝑘 moment curve which follows (essentially) from the Newton-Girard identities. The iteration is surprisingly efficient when it controls 𝑙 2 𝐿 𝑝 decoupling by 𝑙 2 𝐿 𝑝-2𝑘 decoupling as long as both 𝑝 and 𝑝 -2𝑘 are supercritical. However, after about 1 2𝑘 (𝑝 -𝑘(𝑘+1) 2 ) steps, we enter the subcritical regime for which the iteration becomes inefficient and this is why we accrue an additional 𝛿 - 𝑘 2 2𝑝 (1-1 𝑘 ) 𝑝∕(2𝑘) term. When 𝑘 = 2, the argument for Corollary 1.2 uses 𝑂(𝑝) steps to prove a weak nonsharp 𝑙 2 𝐿 𝑝 decoupling estimate. This is to be compared to the modern proof of decoupling for the parabola where to prove the sharp critical 𝑙 2 𝐿 6 decoupling, one uses 𝑂(𝜀 -1 ) many steps (see, for example, the proof of [START_REF] Li | An 𝑙 2 decoupling interpretation of efficient congruencing: the parabola[END_REF]Lemma 2.12]). In the harmonic analysis literature, iterating on 𝑝 is not a new idea as such an argument was already used by Drury [START_REF] Drury | Restrictions of Fourier transforms to curves[END_REF] to prove cubic moment curve restriction, though we believe this is the first time such an argument has appeared in the decoupling literature.

See also [START_REF]Diameter free estimates for the quadratic Vinogradov mean value theorem[END_REF] by the fourth author for a similar idea in the additive combinatorics literature that was recently used to obtain diameter free estimates for the quadratic VMVT. Additionally, at each iterative step, three scales are key: the smallest scale 𝛿, the intermediate scale 𝛿 1∕𝑘 , and the largest scale 1 (though strictly speaking in our proof the largest scale is actually 𝛿 𝜀 rather than 1 for technical reasons). This can be compared to [START_REF] Bourgain | Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three[END_REF][START_REF] Guo | A short proof of 𝓁 2 decoupling for the moment curve[END_REF] which uses scales 𝛿, 𝛿 𝜀 and 1.

This paper is organized as follows: In Section 2, we review some basic geometric and harmonic analysis facts in ℚ 𝑞 that will be used throughout this paper. In Section 3, we review the refinement of the 1973 argument of Karatsuba at a high level. In Section 4, we prove Lemma 4.2 which is the main lemma that is used to prove Theorem 1.1. This is accomplished via combining a standard broad-narrow argument in Subsection 4.1.1 and some geometric properties of the moment curve that use the Newton-Girard identites, see Lemma 4.4. In Section 5, we dyadically pigeonhole to obtain some uniformity in our estimates and prove Theorem 1.1 and Corollary 1.2. Finally, in the Appendix, we include a proof of 𝔇 2𝑘 (𝛿) ≲ 𝜀 𝛿 -𝜀 for completeness.

WAVEPACKET DECOMPOSITION AND SOME BASIC GEOMETRIC FACTS

Throughout this paper, we will make use of wavepacket decomposition which allows us to decompose a function 𝑓, which is Fourier supported in some 𝜃 𝐾 , into linear combinations of indicator functions of translates of the parallelepiped "dual" to 𝜃 𝐾 . That the 𝑞-adic character 𝜒 is trivial on ℤ 𝑞 gives a much cleaner wavepacket decomposition when working over ℚ 𝑞 than over ℝ. See [30, section 3] or [17, section 2.4] for some discussion about wavepacket decomposition over ℝ in the context of the paraboloid (though the same ideas apply for the degree 𝑘 moment curve).

Fix 𝛿 ∈ 𝑞 -ℕ . It will be convenient to introduce the shorthand

𝜃 𝛿 ∶= 𝛿ℤ 𝑞 × 𝛿 2 ℤ 𝑞 × ⋯ × 𝛿 𝑘 ℤ 𝑞
and

𝑇 𝛿 ∶= 𝛿 -1 ℤ 𝑞 × 𝛿 -2 ℤ 𝑞 × ⋯ × 𝛿 -𝑘 ℤ 𝑞 .
They are dual to each other in the sense that

𝑇 𝛿 = {𝑥 ∈ ℚ 𝑘 𝑞 ∶ |𝑥 ⋅ 𝜉| ⩽ 1 for all 𝜉 ∈ 𝜃 𝛿 }.
Since for any 1 ⩽ 𝑗 ⩽ 𝑘, any interval in ℚ 𝑞 of length 𝛿 𝑗 is the disjoint union of 𝛿 -(𝑘-𝑗) many intervals of length 𝛿 𝑘 , it follows that 𝜃 𝛿 is the disjoint union of 𝛿 -𝑘(𝑘-1) 2 many cubes of side lengths 𝛿 𝑘 in ℚ 𝑘 𝑞 . Similarly, any cube in ℚ 𝑘 𝑞 of side length 𝛿 -𝑘 is a disjoint union of 𝛿 -𝑘(𝑘-1) 2 many translates of 𝑇 𝛿 . Now for 𝑎 ∈ ℤ 𝑞 , let 𝑀 𝑎 be the 𝑘 × 𝑘 lower-triangular matrix given by 𝑀 𝑎 = (𝛾 ′ (𝑎) 𝛾 ′′ (𝑎) ⋯ 𝛾 (𝑘) (𝑎))

where we view 𝛾 (𝑗) (𝑎) as a column vector. Then for any 𝐾 ∈ 𝑃 𝛿 , we have

𝜃 𝐾 = 𝛾(𝑎) + 𝑀 𝑎 𝜃 𝛿 (2.1)
for any 𝑎 ∈ 𝐾. In fact, the right-hand side is independent of 𝑎 ∈ 𝐾 because if 𝑏 ∈ 𝐾, then

𝛾(𝑏) = 𝛾(𝑎) + 𝑘 ∑ 𝑗=1 (𝑗!) -1 𝛾 (𝑗) (𝑎)(𝑏 -𝑎) 𝑗 ∈ 𝛾(𝑎) + 𝑀 𝑎 𝜃 𝛿 ,
and where the second matrix on the right-hand side preserves 𝜃 𝛿 = 𝛿ℤ 𝑞 × 𝛿 2 ℤ 𝑞 × ⋯ × 𝛿 𝑘 ℤ 𝑞 (here we have used the fact that |𝑘!| = 1 in ℚ 𝑞 because 𝑞 > 𝑘).

𝑀 𝑎 = 𝑀 𝑏 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 … 0 (1!) -1 (𝑏 -𝑎) 1 … 0 (2!) -1 (𝑏 -𝑎) 2 (1!) -1 (𝑏 -𝑎) … 0 ⋮ ⋱ ((𝑘 -1)!) -1 (𝑏 -𝑎) 𝑘-1 ((𝑘 -2)!) -1 (𝑏 -𝑎) 𝑘-2 … 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ( 2
For 𝐾 ∈ 𝑃 𝛿 and any 𝑎 ∈ 𝐾, let 𝑇 0,𝐾 be the dual parallelepiped to 𝜃 𝐾 centered at the origin given by 𝑇 0,𝐾 = {𝑥 ∈ ℚ 𝑘 𝑞 ∶ |𝑥 ⋅ (𝜉 -𝛾(𝑎))| ⩽ 1 for all 𝜉 ∈ 𝜃 𝐾 }.

Using (2.1), it is not hard to see that

𝑇 0,𝐾 = {𝑥 ∈ ℚ 𝑘 𝑞 ∶ |𝑥 ⋅ 𝛾 (𝑗) (𝑎)| ⩽ 𝛿 -𝑗 for all 1 ⩽ 𝑗 ⩽ 𝑘} = {𝑥 ∈ ℚ 𝑘 𝑞 ∶ 𝑀 𝑇 𝑎 𝑥 ∈ 𝑇 𝛿 } = 𝑀 -𝑇 𝑎 𝑇 𝛿
for any 𝑎 ∈ 𝐾. This parallelepiped depends only on 𝐾 but not on the choice of 𝑎 ∈ 𝐾, as (2.2) shows that

𝑀 -𝑇 𝑎 = 𝑀 -𝑇 𝑏 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 𝑂(𝛿) 𝑂(𝛿 2 ) … 𝑂(𝛿 𝑘-1 ) 0 1 𝑂(𝛿) … 𝑂(𝛿 𝑘-2 ) 0 0 1 … 𝑂(𝛿 𝑘-3 ) ⋮ ⋱ 0 0 0 … 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, where 𝑂(𝛿 𝑗 ) is some number in ℚ 𝑞 with norm ⩽ 𝛿 𝑗 , and the second matrix on the right-hand side is a bijection that preserves 𝑇 𝛿 by the ultrametric inequality.

Lemma 2.1. Let 𝛿 ∈ 𝑞 -ℕ and fix 𝐾 ∈ 𝑃 𝛿 . Then

(i) 𝜃 𝐾 -𝜃 𝐾 is the disjoint union of 𝛿 -𝑘(𝑘-1)
2 cubes of side lengths 𝛿 𝑘 , and

(ii) every cube of side length 𝛿 -𝑘 in ℚ 𝑘 𝑞 is the disjoint union of 𝛿 -𝑘(𝑘-1) 2 many translates of 𝑇 0,𝐾 .

Proof.

(i) Recall that 𝜃 𝛿 is the disjoint union of 𝛿 -𝑘(𝑘-1) 2 cubes of side lengths 𝛿 𝑘 . As 𝑀 𝑎 is a bijection that maps cubes of side length 𝛿 𝑘 to cubes of side length 𝛿 𝑘 for any 𝑎 ∈ 𝐾, and 𝜃 𝐾 -𝜃 𝐾 = 𝑀 𝑎 𝜃 𝛿 for any 𝑎 ∈ 𝐾, the assertion follows. Note that 𝜃 𝐾 -𝜃 𝐾 is just a translation of 𝜃 𝐾 to the origin.

(ii) Recall that any cube in ℚ 𝑘 𝑞 of side length 𝛿 -𝑘 is a disjoint union of 𝛿 -𝑘(𝑘-1) 2 many translates of 𝑇 𝛿 . As 𝑀 -𝑇 𝑎 is a bijection that maps cubes of side length 𝛿 -𝑘 to cubes of side length 𝛿 -𝑘 for any 𝑎 ∈ 𝐾, and 𝑇 0,𝐾 = 𝑀 -𝑇 𝑎 𝑇 𝛿 for any 𝑎 ∈ 𝐾, the assertion follows. □ From Lemma 2.1(ii), we may deduce that translates of 𝑇 0,𝐾 tile ℚ 𝑘 𝑞 ; we denote the collection of such translates by 𝕋(𝐾). We are now ready to state the version of wavepacket decomposition that we will use.

Lemma 2.2 (Wavepacket decomposition).

Let 𝛿 ∈ 𝑞 -ℕ and fix 𝐾 ∈ 𝑃 𝛿 . Let g be a Schwartz function with Fourier transform supported in 𝜃 𝐾 . Then |g| is constant on every 𝑇 ∈ 𝕋(𝐾), and ĝ1 𝑇 is supported 20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License on 𝜃 𝐾 for every 𝑇 ∈ 𝕋(𝐾). Hence, it is natural to write

g = ∑ 𝑇∈𝕋(𝐾) g1 𝑇 , (2.3)
where each term g1 𝑇 (which we will call a "wavepacket") is Fourier supported on 𝜃 𝐾 and has constant modulus on every 𝑇 ∈ 𝕋(𝐾). It also follows that if  is any subset of 𝕋(𝐾), then ∑ 𝑇∈ g1 𝑇 is Fourier supported in 𝜃 𝐾 .

Proof. First, to prove that |g| is constant on any translates of 𝑇 0,𝐾 , one only needs to prove the case when 𝛿 = 1, 𝐾 = ℤ 𝑞 , and then apply a change of variables, but we opt for a more explicit proof. We will show that |g(𝑥)| is constant for all 𝑥 ∈ 𝐴 + 𝑇 0,𝐾 for any 𝐴 ∈ ℚ 𝑘 𝑞 . By Fourier inversion we have that

|g(𝑥)| = | ∫ 𝜃 𝐾 ĝ(𝜉)𝜒(𝜉 ⋅ 𝑥) 𝑑𝜉| = | ∫ |𝑡 1 |⩽𝛿,…,|𝑡 𝑘 |⩽𝛿 𝑘 ĝ( 𝛾(𝑎) + 𝑘 ∑ 𝑗=1 𝑡 𝑗 𝛾 (𝑗) (𝑎) ) 𝜒 ([ 𝛾(𝑎) + 𝑘 ∑ 𝑗=1 𝑡 𝑗 𝛾 (𝑗) (𝑎) ] ⋅ 𝑥 ) 𝑑𝑡| = | ∫ |𝑡 1 |⩽𝛿,…,|𝑡 𝑘 |⩽𝛿 𝑘 ĝ(𝛾(𝑎) + 𝑀 𝑎 𝑡)𝜒(𝑀 𝑇 𝑎 𝑥 ⋅ 𝑡) 𝑑𝑡|.
For 𝑥 ∈ 𝐴 + 𝑇 0,𝐾 , we write

𝑥 = 𝐴 + 𝑀 -𝑇 𝑎 𝑦 ′ where |𝑦 ′ 𝑗 | ⩽ 𝛿 -𝑗 for 𝑗 = 1, 2, … , 𝑘. Therefore, |g(𝑥)| = | ∫ |𝑡 1 |⩽𝛿,…,|𝑡 𝑘 |⩽𝛿 𝑘 ĝ(𝛾(𝑎) + 𝑀 𝑎 𝑡)𝜒(𝑀 𝑇 𝑎 𝐴 ⋅ 𝑡)𝜒(𝑦 ′ ⋅ 𝑡) 𝑑𝑡| = | ∫ |𝑡 1 |⩽𝛿,…,|𝑡 𝑘 |⩽𝛿 𝑘 ĝ(𝛾(𝑎) + 𝑀 𝑎 𝑡)𝜒(𝑀 𝑇 𝑎 𝐴 ⋅ 𝑡) 𝑑𝑡|,
where we have used that 𝑦 ′ ⋅ 𝑡 ∈ ℤ 𝑞 , and so 𝜒(𝑦 ′ ⋅ 𝑡) = 1. The right-hand side is then independent of 𝑦 ′ and so the above equality is true for all 𝑥 ∈ 𝐴 + 𝑇 0,𝐾 . In particular, this shows that |g| is a constant on 𝐴 + 𝑇 0,𝐾 . This constant depends on 𝐾, g and 𝐴, but is a constant nonetheless.

Next, to prove that ĝ1 𝑇 is supported on 𝜃 𝐾 , it suffices to observe that ĝ1 𝑇 = ĝ * 1𝑇 , and that 1𝑇 is supported on 𝜃 𝐾 -𝜃 𝐾 for every 𝑇 ∈ 𝕋(𝐾): in fact, for every 𝑇 ∈ 𝕋(𝐾), 1𝑇 is a modulation of 1𝑇 0,𝐾 , and if 𝑎 is any point in 𝐾, then 𝑇 0,𝐾 = 𝑀 -𝑇 𝑎 𝑇 𝛿 . It follows that

1𝑇 0,𝐾 (𝜉) = ∫ 𝑀 -𝑇 𝑎 𝑇 𝛿 𝜒(-𝑥 ⋅ 𝜉)𝑑𝑥 = det(𝑀 𝑎 ) -1 ∫ 𝑇 𝛿 𝜒(-𝑀 -𝑇 𝑎 𝑦 ⋅ 𝜉)𝑑𝑦 = det(𝑀 𝑎 ) -1 𝛿 -𝑘(𝑘+1)∕2 1 𝜃 𝛿 (𝑀 -1 𝑎 𝜉)
is supported on 𝑀 𝑎 𝜃 𝛿 = 𝜃 𝐾 -𝜃 𝐾 . Finally, the decomposition (2.3) follows because parallelepipeds in 𝕋(𝐾) tile ℚ 𝑘 𝑞 . This completes the proof of the lemma. □

SKETCH OF THE KARATSUBA ARGUMENT

Before we dive into the proof of Theorem 1.1, we review the proof of (1.3) with an eye toward interpreting each step into decoupling language. See also, for example, [31, section 5.1] or [29, Theorem 13 -Lemma 21] for more details of the number theoretic argument. Just for this section, we revert back to calling 𝑝 a prime so as to best match these references.

3.1

Step 1: Introducing some 𝒑-adic separation

Given 𝑋 ⩾ 1, one finds, using the Prime Number Theorem, a prime 𝑝 ∼ 𝑋 1∕𝑘 such that 𝐽 𝑠,𝑘 (𝑋) is controlled by 𝐽 𝑠,𝑘 (𝑋, 𝑝), where 𝐽 𝑠,𝑘 (𝑋, 𝑝) is defined to be the number of solutions

(𝑥 1 , … , 𝑥 𝑠 , 𝑦 1 , … , 𝑦 𝑠 ) ∈ ([1, 𝑋] ∩ ℕ) 2𝑠 to (1.1)
with the additional condition that 𝑥 1 , … , 𝑥 𝑘 are pairwise distinct mod 𝑝 and 𝑦 1 , … , 𝑦 𝑘 are pairwise distinct mod 𝑝. As 𝑝 is rather large, this is a rather mild condition and so we heuristically should still expect 𝐽 𝑠,𝑘 (𝑋) ≈ 𝐽 𝑠,𝑘 (𝑋, 𝑝). The benefit of this extra 𝑝-adic separation (transversality) in these 2𝑘 variables is that we will get to apply Linnik's lemma (in Step 3, (3.3)) which will up to permutation uniquely determine these variables.

Step 2: Applying the union bound/Hölder

We now write 𝐽 𝑠,𝑘 (𝑋, 𝑝) as

∫ [0,1] 2𝑠 | ∑ 𝑎 1 ,…,𝑎 𝑘 (mod 𝑝) 𝑎 𝑖 pairwise distinct 𝑘 ∏ 𝑗=1 ∑ 𝑛 𝑗 ≡𝑎 𝑗 (mod 𝑝) 1⩽𝑛 𝑗 ⩽𝑋 𝑒 ( 𝑛 𝑗 𝛼 1 + ⋯ + 𝑛 𝑘 𝑗 𝛼 𝑘 ) | 2 | ∑ 1⩽𝑛⩽𝑋 𝑒 ( 𝑛𝛼 1 + ⋯ + 𝑛 𝑘 𝛼 𝑘 ) | 2𝑠-2𝑘 𝑑𝛼. Write | ∑ 1⩽𝑛⩽𝑋 | 2𝑠-2𝑘 = | ∑ 𝑎 (mod 𝑝) ∑ 𝑛≡𝑎 (mod 𝑝) | 2𝑠-2𝑘
and apply Hölder's inequality to control the above by

𝑝 2𝑠-2𝑘 max 𝑎 (mod 𝑝) ∫ [0,1] 2𝑠 | | | ∑ 𝑎 1 ,…,𝑎 𝑘 (mod 𝑝) 𝑎 𝑖 pairwise distinct 𝑘 ∏ 𝑗=1 ∑ 𝑛 𝑗 ≡𝑎 𝑗 (mod 𝑝) 1⩽𝑛 𝑗 ⩽𝑋 𝑒(𝑛 𝑗 𝛼 1 + ⋯ + 𝑛 𝑘 𝑗 𝛼 𝑘 )| 2 × | ∑ 𝑛≡𝑎 (mod 𝑝) 1⩽𝑛⩽𝑋 𝑒(𝑛𝛼 1 + ⋯ + 𝑛 𝑘 𝛼 𝑘 )| 2𝑠-2𝑘 𝑑𝛼. (3.1)
Denote the integral above to be 𝐽 𝑠,𝑘 (𝑋, 𝑝, 𝑎). This expression counts the number of solutions

(𝑥 1 , … , 𝑥 𝑠 , 𝑦 1 , … , 𝑦 𝑠 ) ∈ ([1, 𝑋] ∩ ℕ) 2𝑠 to (1.1) with 𝑥 1 , … , 𝑥 𝑘 pairwise distinct mod 𝑝, 𝑦 1 , … , 𝑦 𝑘 pairwise distinct mod 𝑝, and 𝑥 𝑘+1 ≡ ⋯ ≡ 𝑥 𝑠 ≡ 𝑦 𝑘+1 ≡ ⋯ ≡ 𝑦 𝑠 ≡ 𝑎 (mod 𝑝).

Step 3: Solution counting

Translation invariance of the Vinogradov system implies that we may bound 𝐽 𝑠,𝑘 (𝑋, 𝑝, 𝑎) by 𝐽 𝑠,𝑘 (𝑋, 𝑝, 0). Rearrange the Vinogradov system (1.1) as

𝑥 𝑗 𝑘+1 + ⋯ + 𝑥 𝑗 𝑠 -𝑦 𝑗 𝑘+1 -⋯ -𝑦 𝑗 𝑠 = 𝑦 𝑗 1 + ⋯ + 𝑦 𝑗 𝑘 -𝑥 𝑗 1 -⋯ -𝑥 𝑗 𝑘 , 1 ⩽ 𝑗 ⩽ 𝑘, (3.2
𝑥 𝑗 1 + ⋯ + 𝑥 𝑗 𝑘 ≡ 𝐻 𝑗 (mod 𝑝 𝑗 ), 1 ⩽ 𝑗 ⩽ 𝑘,
where the 𝑥 𝑖 are pairwise disjoint mod 𝑝 for some 𝐻 𝑗 that depends on (𝑦 1 , … , 𝑦 𝑘 ) (of which there are ⩽ 𝑋 𝑘 many possibilities) and (𝑥 𝑘+1 , … , 𝑥 𝑠 , 𝑦 𝑘+1 , … , 𝑦 𝑠 ) (of which there are ⩽ 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝) many possibilities). As 𝑝 𝑘 ∼ 𝑋, instead of counting integers between 1 and 𝑋, we can count the 𝑥 𝑖 mod 𝑝 𝑘 . Thus, it remains to count the number of residue classes (𝑥 1 (mod 𝑝 𝑘 ), … , 𝑥 𝑘 (mod 𝑝 𝑘 )) such that

𝑥 𝑗 1 + ⋯ + 𝑥 𝑗 𝑘 ≡ 𝐻 𝑗 (mod 𝑝 𝑗 ), 1 ⩽ 𝑗 ⩽ 𝑘 (3.3)
and 𝑥 𝑖 (mod 𝑝 𝑘 ) are pairwise distinct mod 𝑝. Linnik's lemma [START_REF] Linnik | On Weyl's sums[END_REF] then says that there are at most 𝑘!𝑝 𝑘(𝑘-1)∕2 many such 𝑘-tuples of residue classes and the proof follows from first upgrading all residue classes mod 𝑝 𝑗 in (3.3) to mod 𝑝 𝑘 (by paying a cost of 𝑝 𝑘(𝑘-1)∕2 ) and then using the Newton-Girard identities that essentially uniquely determine the 𝑥 1 , … , 𝑥 𝑘 (up to permutation). This bound is efficient because probabilistic heuristics suggest that we should expect ≈ (𝑝 𝑘 ) 𝑘 ∕𝑝 𝑘(𝑘+1)∕2 = 𝑝 𝑘(𝑘-1)∕2 many solutions. Thus, we have that 𝐽 𝑠,𝑘 (𝑋, 𝑝, 0) ≲ 𝑘 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝)𝑋 𝑘 𝑝 𝑘(𝑘-1)∕2 .

(3.4)

Step 4: Iteration

Putting Steps 1-3 together, we obtain the iteration that 𝐽 𝑠,𝑘 (𝑋) ≲ 𝑘 𝑝 2𝑠-2𝑘 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝)𝑋 𝑘 𝑝 𝑘(𝑘-1)∕2 .

Running this iteration about 𝑂(𝑠∕𝑘) many steps reduces to an estimate on 𝐽 𝑘,𝑘 (𝑋) from which one can easily compute there are 𝑂(𝑋 𝑘 ) many solutions by the Newton-Girard identities. The iteration (3.5) is sharp if both 𝑠 and 𝑠 -𝑘 are supercritical. If they are, then heuristically, we expect 𝐽 𝑠,𝑘 (𝑋) ≈ 𝑋 2𝑠-𝑘(𝑘+1)∕2 and 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝) ≈ (𝑋∕𝑝) 2(𝑠-𝑘)-𝑘(𝑘+1)∕2 . Then the right-hand side of (3.5) becomes 𝑋 2𝑠 𝑋 -3𝑘∕2-𝑘 2 ∕2 𝑝 𝑘 2 which is equal to 𝑋 2𝑠-𝑘(𝑘+1)∕2 because 𝑝 ∼ 𝑋 1∕𝑘 . However, both sides are not the same if one of 𝑠 or 𝑠 -𝑘 is subcritical. This is where the inefficiency of 𝑋 𝑘 2

2 (1-1 𝑘 ) 𝑠∕𝑘 comes from.

Interpreting Steps 1-4 into decoupling

Having briefly summarized the number theoretic argument into four steps, we now briefly sketch the main points to interpret into decoupling. First we discuss the scales needed in the proof.

From Steps 1 and 3, there are three scales: the largest scale 𝑋, the intermediate scale 𝑝 ∼ 𝑋 1∕𝑘 , and the smallest scale 1. Correspondingly in our proof, we use three scales: the smallest scale 𝛿, the intermediate scale 𝜈 ∶= 𝑞 ⌊log 𝑞 𝛿 1∕𝑘 ⌋ ∼ 𝛿 1∕𝑘 , and the largest scale 1. For some technical reasons surrounding the broad-narrow reduction, in lieu of the scale 1, we will actually use the scale 𝜅 ∶= 𝑞 ⌊log 𝑞 𝛿 𝜀 ⌋ where 𝜀 is as in (1.8). Next, we discuss the reduction to the decoupling analogue of (3.1). In Step 1, two residue classes being distinct mod 𝑝 means they are 𝑝-adically separated by a distance 1 and so this should correspond to two intervals that are 1-separated. To get around the use of the Prime Number Theorem, we instead make use of broad-narrow reduction due to Bourgain and Guth in [START_REF] Bourgain | Bounds on oscillatory integral operators based on multilinear estimates[END_REF] which will allow us to reduce to controlling a multilinear decoupling expression.

Third, the loss of 𝑝 2𝑠-2𝑘 in Step 2 above deserves some mention. This loss comes from essentially having applied the union bound

| | | ∑ 1⩽𝑛⩽𝑋 𝑒(𝑛𝛼 1 + ⋯ + 𝑛 𝑘 𝛼 𝑘 ) | | | = | | | ∑ 𝑎 (mod 𝑝) ∑ 𝑛≡𝑎 (mod 𝑝) 1⩽𝑛⩽𝑋 𝑒(𝑛𝛼 1 + ⋯ + 𝑛 𝑘 𝛼 𝑘 ) | | | ⩽ 𝑝 max 𝑎 (mod 𝑝) | | | ∑ 𝑛≡𝑎 (mod 𝑝) 1⩽𝑛⩽𝑋 𝑒(𝑛𝛼 1 + ⋯ + 𝑛 𝑘 𝛼 𝑘 ) | | | .
Heuristically, we expect this inequality to be efficient because each ∑ 𝑛≡𝑎 (mod 𝑝) contributes equally to the entire sum as the exponential sum should not bias one residue class mod 𝑝 over another. This, however, is not necessarily true in the decoupling case and will require us to obtain some extra uniformity via dyadic pigeonholing, see Subsection 5.1, later.

Finally, to interpret the solution counting Step 3, we make use of the simple identity

∫ ℚ 𝑘 𝑞 𝑓(𝑥) 𝑑𝑥 = f(0),
which converts the integral of 𝑓 into a question of whether 0 is contained in the support of f. This is done in Lemma 4.4 and the proof relies on the Newton-Girard identities, much like in the proof of Linnik's lemma. This part of the argument requires that 𝑝 is even and is reminiscent of a Córdoba-Fefferman argument (see, for example, [8, section 3.2] or [START_REF] Córdoba | The Kakeya maximal function and the spherical summation multipliers[END_REF][START_REF] Córdoba | Geometric Fourier analysis[END_REF][START_REF] Fefferman | A note on spherical summation multipliers[END_REF]).

THE MAIN LEMMA

One standard property about the moment curve decoupling constant that we use is affine rescaling. This property plays the analogue of translation-dilation invariance of the

) 𝑝∕2 + 𝐶𝑞 -𝑘(𝑘-1) 𝜅 -(𝑘 2 +4𝑘-2) 𝜈 -𝑘(𝑘-1)∕2 𝑁 𝑝-2𝑘 × 𝔇 𝑝-2𝑘 ( 𝛿 𝜈 ) 𝑝-2𝑘 max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 max 𝐽∈𝑃 𝜈 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾 ′ ∈𝑃 𝛿 (𝐽) ‖g 𝐾 ′ ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ (𝑝-2𝑘)∕2
, where 𝑁 is the number of 𝐽 ∈ 𝑃 𝜈 for which g 𝐽 ≠ 0 and 𝐶 depends only on 𝑘 and 𝑝.

Here 𝜅 is a somewhat technical parameter that is chosen to be roughly 𝛿 𝜀 later in Section 5. However, on a first reading, it might be more convenient for the reader to take 𝜅 = 1∕𝑞 to better grasp the moving parts of the argument. The somewhat nonstandard decoupling right-hand side in Lemma 4.2 is reminiscent of the right-hand side used in [START_REF] Guth | Improved decoupling for the parabola[END_REF]Theorem 1.2]. To give more context to the above lemma, the following estimate is true: Lemma 4.3. For any 𝑝 > 2𝑘, we have

( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ) 𝑝∕2 ⩽ 𝑁 (𝑝-2𝑘)∕2 max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 max 𝐽∈𝑃 𝜅 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 (𝐽) ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ (𝑝-2𝑘)∕2
, where 𝑁 is as defined in Lemma 4.2.

Proof. Hölder's inequality gives us

‖g 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽ ‖g 𝐾 ‖ 2𝑘 𝑝 𝐿 ∞ (ℚ 𝑘 𝑞 ) ‖g 𝐾 ‖ 1-2𝑘 𝑝 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 )
, and so, applying

( ∑ 𝐾 𝑎 2𝑘 𝑝 𝐾 𝑏 2𝑘 𝑝 𝐾 𝑐 2(1-2𝑘 𝑝 ) 𝐾 ) 𝑝 2 ⩽ (max 𝐾 𝑎 𝐾 ) 𝑘 ( ∑ 𝐾 𝑏 𝐾 ) 𝑘 ( ∑ 𝐾 𝑐 2 𝐾 ) 𝑝-2𝑘
) 𝑝∕2 ⩽ max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 ( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 )
) (𝑝-2𝑘)∕2

.

It remains to observe that

( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ) (𝑝-2𝑘)∕2 ⩽ 𝑁 (𝑝-2𝑘)∕2 max 𝐽∈𝑃 𝜅 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 (𝐽) ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ (𝑝-2𝑘)∕2
.

□

Suppose for a moment that in Lemma 4.3, we had an equality instead of an inequality. This is indeed the case when g(𝑥) is the exponential sum 𝑋 -100𝑘 2 1 |𝑥|⩽𝑋 100𝑘 ∑ 𝑋 𝑗=1 𝑒(𝛾(𝑗) ⋅ 𝑥) that arises in using decoupling to estimate the number of solutions in (1.3). As 𝑁 ⩽ 𝜈 -1 (and taking, for convenience, 𝜅 = 1∕𝑞), Lemma 4.2 would give us

𝔇 𝑝 (𝛿) 𝑝 ⩽ 𝐶 𝔇 𝑝 (𝑞𝛿) 𝑝 + 𝐶𝑞 5𝑘-2 𝜈 -𝑝 2 +𝑘-𝑘(𝑘-1) 2 𝔇 𝑝-2𝑘 ( 𝛿 𝜈 ) 𝑝-2𝑘 (4.1)
Heuristically, we expect this iteration to be efficient as long as 𝑝 -2𝑘 (and so also 𝑝) is supercritical. To see this, if 𝑟 is supercritical, then we heuristically expect that 𝔇 𝑟 (𝛿) 𝑟 ≈ 𝛿 -𝑟 2 + 𝑘(𝑘+1) 2 for all 𝛿. Thus, the iteration should be efficient if with this assumption on the size of 𝔇 𝑟 (𝛿) 𝑟 , both sides of (4.1) are the same. The right-hand side of (4.1) is then

∼ 𝑞 ( 𝛿 -1 𝑘 ) 𝑝 2 +𝑘-𝑘(𝑘-1) 2 ( 𝛿 -1+ 1 𝑘 ) 𝑝-2𝑘 2 -𝑘(𝑘+1) 2 = 𝛿 - ( 𝑝 2 -𝑘(𝑘+1)
2

)
, which is comparable to the left-hand side of (4.1). A similar calculation shows that this iteration is not efficient if at least one of 𝑝 or 𝑝 -2𝑘 is subcritical.

Unfortunately the reverse inequality in Lemma 4.3 fails to hold for general g. This is because we lack the uniformity in the exponential sum that one considers when one counts solutions to the Vinogradov system. This uniformity can be restored by pigeonholing, which only produces 𝛿 -𝜀 losses. This pigeonholing must be done before one applies induction on scales and iterates on the Lebesgue exponent 𝑝. The full argument is carried out in detail in Section 5.

Proof of Lemma 4.2

The proof of Lemma 4.2 uses a broad/narrow dichotomy, due to Bourgain and Guth [START_REF] Bourgain | Bounds on oscillatory integral operators based on multilinear estimates[END_REF] combined with some basic geometric geometric properties of the moment curve. See also, for example, [8, chapter 7].

4.1.1

The broad-narrow argument

(𝑥)| ⩽ 𝜅 -1 max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)| ⩽ 𝜅 -1 𝜅 -(𝑘-1)∕𝑘 |g 𝐼 ′ 1 (𝑥) … g 𝐼 ′ 𝑘 (𝑥)| 1∕𝑘 ⩽ 𝜅 -2+1∕𝑘 max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 |g 𝐼 1 (𝑥) … g 𝐼 𝑘 (𝑥)| 1∕𝑘 .
Here we used that in ℚ 𝑘 𝑞 , two distinct intervals of the same length are separated by at least that length. Alternatively,  𝑥 contains at most 𝑘 -1 intervals, in which case

|g(𝑥)| ⩽ ∑ 𝐼∈ 𝑥 |g 𝐼 (𝑥)| + ∑ 𝐼∈𝑃 𝜅 ⧵ 𝑥 |g 𝐼 (𝑥)| < (𝑘 -1) max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)| + ∑ 𝐼∈𝑃 𝜅 ⧵ 𝑥 𝜅 max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)| < 𝑘 max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)|.
As a result, we obtain the pointwise bound that for each 𝑥 ∈ ℚ 𝑘 𝑞 , we have

|g(𝑥)| ⩽ 𝑘 max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)| + 𝜅 -2+1∕𝑘 max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 |g 𝐼 1 (𝑥) … g 𝐼 𝑘 (𝑥)| 1∕𝑘
which, upon raising both sides to power 2𝑘 and applying (𝐴 + 𝐵) 2𝑘 ⩽ 2 2𝑘-1 (𝐴 2𝑘 + 𝐵 2𝑘 ) (a consequence of the convexity of 𝑥 ↦ 𝑥 2𝑘 ), yields

|g(𝑥)| 2𝑘 ⩽ 2 2𝑘-1 𝑘 2𝑘 max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)| 2𝑘 + 2 2𝑘-1 𝜅 -(4𝑘-2) max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 |g 𝐼 1 (𝑥) … g 𝐼 𝑘 (𝑥)| 2 . (4.2)
Using this pointwise bound while integrating we find that for some 𝐶 depending on 𝑘. Hölder's inequality followed by Minkowski's inequality implies that the first term satisfies

∫ ℚ 𝑘 𝑞 |g| 𝑝 = ∫ ℚ 𝑘 𝑞 |g| 2𝑘 |g| 𝑝-2𝑘 ⩽ 𝐶 ∫ ℚ 𝑘 𝑞 (max 𝐼∈𝑃 𝜅 |g 𝐼 | 2 ) 𝑘 |g| 𝑝-2𝑘 + 𝐶𝜅 -(4𝑘-2) ∫ ℚ 𝑘 𝑞 max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g| 𝑝-2𝑘 ⩽ 𝐶 ∫ ℚ 𝑘 𝑞 ( ∑ 𝐼∈𝑃 𝜅 |g 𝐼 | 2 ) 𝑘 |g| 𝑝-2𝑘 + 𝐶𝜅 -(4𝑘-2) ∑ 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g| 𝑝-2𝑘 ⩽ 𝐶 ∫ ℚ 𝑘 𝑞 ( ∑ 𝐼∈𝑃 𝜅 |g 𝐼 | 2 ) 𝑘 |g| 𝑝-2𝑘 + 𝐶𝜅 -(4𝑘-
𝐶 ∫ ℚ 𝑘 𝑞 ( ∑ 𝐼∈𝑃 𝜅 |g 𝐼 | 2 ) 𝑘 |g| 𝑝-2𝑘 ⩽ 𝐶 ⎛ ⎜ ⎜ ⎝ ∫ ℚ 𝑘 𝑞 ( ∑ 𝐼∈𝑃 𝜅 |g 𝐼 | 2 ) 𝑝∕2 ⎞ ⎟ ⎟ ⎠ 2𝑘∕𝑝 ( ∫ ℚ 𝑘 𝑞 |g| 𝑝 ) (𝑝-2𝑘)∕𝑝 ⩽ 𝐶 ( ∑ 𝐼∈𝑃 𝜅 ‖g 𝐼 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )
) 𝑘 (

∫ ℚ 𝑘 𝑞 |g| 𝑝 ) (𝑝-2𝑘)∕𝑝 ⩽ 1 2 ∫ ℚ 𝑘 𝑞 |g| 𝑝 + 𝐶 ′ ( ∑ 𝐼∈𝑃 𝜅 ‖g 𝐼 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )
) 𝑝∕2

for some 𝐶 ′ that depends on 𝑘 and 𝑝. The last inequality uses Young's inequality and the fact that 𝑝 ⩾ 2𝑘. Therefore,

∫ ℚ 𝑘 𝑞 |g| 𝑝 ≲ 𝑝 ( ∑ 𝐼∈𝑃 𝜅 ‖g 𝐼 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ) 𝑝∕2 + 𝜅 -(5𝑘-2) max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g| 𝑝-2𝑘 .
Using affine rescaling (Lemma 4.1) and applying the definition (1.5) of our decoupling constant, we deduce that

( ∑ 𝐼∈𝑃 𝜅 ‖g 𝐼 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ) 𝑝∕2 ⩽ 𝔇 𝑝 ( 𝛿 𝜅 ) 𝑝 ( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )
) 𝑝∕2

.

Plugging this into the above yields

∫ ℚ 𝑘 𝑞 |g| 𝑝 ≲ 𝑝 𝔇 𝑝 ( 𝛿 𝜅 ) 𝑝 ( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ) 𝑝∕2 + 𝜅 -(5𝑘-2) max 𝐼 1 ,
) 𝑝∕2 + 𝜅 -(5𝑘-2) 𝑁 𝑝-2𝑘-1 max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 ∑ 𝐽∈𝑃 𝜈 ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐽 | 𝑝-2𝑘 ≲ 𝑝 𝔇 𝑝 ( 𝛿 𝜅 ) 𝑝 ( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ) 𝑝∕2 + 𝜅 -(5𝑘-2) 𝑁 𝑝-2𝑘 max 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 max 𝐽∈𝑃 𝜈 ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐽 | 𝑝-2𝑘 (4.4)
which is the analogue of Step 2 in Subsection 3.2.

To analyze the second term in (4.4), we fix 𝐼 1 , … , 𝐼 𝑘 ∈ 𝑃 𝜅 with 𝑑(𝐼 𝑖 , 𝐼 𝑗 ) > 𝜅 for all 𝑖 ≠ 𝑗, and fix 𝐽 ∈ 𝑃 𝜈 with g 𝐽 ≠ 0. To estimate the integral

∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐽 | 𝑝-2𝑘 , first note that the Fourier transform of |g 𝐽 | 2 = g 𝐽 g 𝐽 is supported in the parallelepiped 𝜃 𝐽 -𝜃 𝐽 , of dimension 𝜈 × 𝜈 2 × ⋯ × 𝜈 𝑘 .
As our hypothesis guarantees that 𝑝 -2𝑘 is an even positive integer, the same is true for the Fourier transform of |g 𝐽 | 𝑝-2𝑘 . Lemma 2.1(i) applied to 𝐽 ∈ 𝑃 𝜈 instead of 𝐾 ∈ 𝑃 𝛿 shows that the Fourier support of |g 𝐽 | 𝑝-2𝑘 is the disjoint union of 𝜈 -𝑘(𝑘-1)∕2 many cubes of side lengths 𝜈 𝑘 , and we denote this collection of cubes by {□}. This corresponds to the fact that we have a 𝑘-tuple of residue classes (𝐻 1 (mod 𝑝), 𝐻 2 (mod 𝑝 2 ), … , 𝐻 𝑘 (mod 𝑝 𝑘 )) that we can upgrade to 𝑝 𝑘(𝑘-1)∕2 many 𝑘-tuples of the form (𝐻 ′ 1 (mod 𝑝 𝑘 ), 𝐻 ′ 2 (mod 𝑝 𝑘 ), … , 𝐻 ′ 𝑘 (mod 𝑝 𝑘 )). Note that the side length 𝜈 𝑘 of the cubes □ is ⩽ 𝛿.

We now apply Fourier inversion and turn products into convolutions. We have

∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐽 | 𝑝-2𝑘 = ∑ 𝐾 𝑖 ∈𝑃 𝛿 (𝐼 𝑖 ) 𝑖=1,…,𝑘 ∑ K𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑗=1,…,𝑘 ∫ ℚ 𝑘 𝑞 g 𝐾 1 … g 𝐾 𝑘 g K1 … g K𝑘 |g 𝐽 | 𝑝-2𝑘 = ∑ □ ∑ 𝐾 𝑖 ∈𝑃 𝛿 (𝐼 𝑖 ) 𝑖=1,…,𝑘 ∑ K𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑗=1,…,𝑘 ĝ𝐾 1 * ⋯ * ĝ𝐾 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 * ( |g 𝐽 | 𝑝-2𝑘 1 □ )(0).
For each fixed □ and K1 ∈ 𝑃 𝛿 (𝐼 1 ), … , K𝑘 ∈ 𝑃 𝛿 (𝐼 𝑘 ), let 𝑆( K1 , … , K𝑘 , □) be the set of all (𝐾 1 , … , 𝐾 𝑘 ) with 𝐾 𝑖 ∈ 𝑃 𝛿 (𝐼 𝑖 ) such that

0 ∈ supp( ĝ𝐾 1 * ⋯ * ĝ𝐾 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 * ( |g 𝐽 | 𝑝-2𝑘 1 □ )). (4.5)
We will prove in Lemma 4.4 that #𝑆( K1 , … , K𝑘 , □) ⩽ (𝑞𝜅) -𝑘(𝑘-1) . If we think of the model case when 𝜅 = 1∕𝑞, this would say that the K𝑖 and □ uniquely determine the 𝐾 𝑖 in (4.5). This is analogous to the situation in Linnik's lemma where once we upgrade (3.3) to residue classes mod 𝑝 𝑘 , the remaining variables are essentially uniquely determined. We now write

∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐽 | 𝑝-2𝑘 = | | | ∑ □ ∑ K𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑗=1,…,𝑘 ⩽ ∑ □ ∑ K𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑗=1,…,𝑘 ∑ (𝐾 1 ,…,𝐾 𝑘 )∈ 𝑆( K1 ,…, K𝑘 ,□) ∫ ℚ 𝑘 𝑞 |g 𝐾 1 … g 𝐾 𝑘 g K1 … g K𝑘 ||g 𝐽 | 𝑝-2𝑘 * |1 □ ⋀ | ⩽ ∑ □ ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 (𝑞𝜅) -𝑘(𝑘-1) max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ∫ ℚ 𝑘 𝑞 |g 𝐽 | 𝑝-2𝑘 * |1 □ ⋀ |. As ∫ ℚ 𝑘 𝑞 |1 □ ⋀ | = 1 and the number of □ is 𝜈 -𝑘(𝑘-1)∕2 , this gives ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐽 | 𝑝-2𝑘 ⩽ 𝜈 -𝑘(𝑘-1) 2 (𝑞𝜅) -𝑘(𝑘-1) max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 ∫ ℚ 𝑘 𝑞 |g 𝐽 | 𝑝-2𝑘 .
Applying affine rescaling shows that this is

⩽ 𝜈 -𝑘(𝑘-1) 2 (𝑞𝜅) -𝑘(𝑘-1) max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) × ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 𝔇 𝑝-2𝑘 ( 𝛿 𝜈 ) 𝑝-2𝑘 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾 ′ ∈𝑃 𝛿 (𝐽) ‖g 𝐾 ′ ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝-2𝑘 2 . (4.6)
One can think of (4.6) as the analogue of (3.4) in Subsection 3.3 in the following way: the term 𝜈 -𝑘(𝑘-1)∕2 (𝑞𝜅) -𝑘(𝑘-1) max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 ∞ plays the role of 𝑝 𝑘(𝑘-1)∕2 from Linnik's lemma, the term ( ∑ K∈𝑃 𝛿 ‖g K ‖ ∞ ) 𝑘 plays the role of 𝑋 𝑘 , and finally the term

𝔇 𝑝-2𝑘 ( 𝛿 𝜈 ) 𝑝-2𝑘 ( ∑ 𝐾 ′ ∈𝑃 𝛿 (𝐽) ‖g 𝐾 ′ ‖ 2 𝑝-2𝑘 ) 𝑝-2𝑘 2
plays the role of the 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝). Plugging (4.6) back to (4.4), we then obtain

∫ ℚ 𝑘 𝑞 |g| 𝑝 ≲ 𝑝 𝔇 𝑝 ( 𝛿 𝜅 ) 𝑝 ( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ) 𝑝∕2 + 𝑞 -𝑘(𝑘-1) 𝜅 -(𝑘 2 +4𝑘-2) 𝜈 -𝑘(𝑘-1) 2 𝑁 𝑝-2𝑘 × 𝔇 𝑝-2𝑘 ( 𝛿 𝜈 ) 𝑝-2𝑘 max 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾 ′ ∈𝑃 𝛿 (𝐽) ‖g 𝐾 ′ ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝-2𝑘 2 .

Geometry of the moment curve

The proof of Lemma 4.2 is now complete modulo the proof of the following lemma, which provides the key geometric input that enables one to count #𝑆( K1 , … , K𝑘 , □). This is the analogue of Linnik's lemma ([29, Corollary 17] and the estimate for 𝐁(𝐠) in the proof of [START_REF] Vaughan | The Hardy-Littlewood method[END_REF]Lemma 5.1]); see also [START_REF] Gressman | Reversing a philosophy: from counting to square functions and decoupling[END_REF]Proposition 1.3] and [2, Proposition 3.1]. Both proofs use the Newton-Girard identities in essentially the same way. The hypothesis that 𝑞 > 𝑘, where 𝑞 is from our base field ℚ 𝑞 and 𝑘 is the degree of the moment curve, plays a role in the following lemma. Then #𝑆( K1 , … , K𝑘 , □) ⩽ (𝑞𝜅) -𝑘(𝑘-1) .

Proof. Assume for the sake of contradiction that #𝑆( K1 , … , K𝑘 , □) > (𝑞𝜅) -𝑘(𝑘-1) ⩾ 1. We can find two 𝑘-tuples of intervals (𝐴 1 , … , 𝐴 𝑘 ) and (𝐵 1 , … , 𝐵 𝑘 ) with each 𝐴 𝑖 , 𝐵 𝑖 ∈ 𝑃 𝛿 (𝐼 𝑖 ) such that

0 ∈ supp( ĝ𝐴 1 * ⋯ * ĝ𝐴 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 * ( |g 𝐽 | 𝑝-2𝑘 1 □ )), (4.7
)

0 ∈ supp( ĝ𝐵 1 * ⋯ * ĝ𝐵 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 * ( |g 𝐽 | 𝑝-2𝑘 1 □ )), (4.8)
and such that there exists an 𝑖 0 with 𝑑(𝐴 𝑖 0 , 𝐵 𝑖 0 ) > (𝑞𝜅) -(𝑘-1) 𝛿. Indeed, if not, picking an arbitrary (𝐶 1 , … , 𝐶 𝑘 ) ∈ 𝑆( K1 , … , K𝑘 , □), shows that any other (𝐷 1 , … , 𝐷 𝑘 ) ∈ 𝑆( K1 , … , K𝑘 , □) must satisfy 𝑑(𝐶 𝑖 , 𝐷 𝑖 ) ⩽ (𝑞𝜅) -(𝑘-1) 𝛿. This gives at most (𝑞𝜅) -𝑘(𝑘-1) many 𝑘-tuples which violates our initial assumption that #𝑆( K1 , … , K𝑘 , □) > (𝑞𝜅) -𝑘(𝑘-1) . Without loss of generality, we may assume that 𝑖 0 = 1.

As for each

(𝑋 -𝑥 1 )(𝑋 -𝑥 2 ) ⋯ (𝑋 -𝑥 𝑘 ) = 𝑘 ∑ 𝑗=0 (-1) 𝑗 𝑒 𝑗 (𝑥 1 , … , 𝑥 𝑘 )𝑋 𝑘-𝑗 (4.11)
and for 𝑗 = 1, 2, … , 𝑘, we have

𝑗𝑒 𝑗 (𝑥 1 , … , 𝑥 𝑘 ) = 𝑗-1 ∑ 𝑖=0 (-1) 𝑖 𝑒 𝑗-𝑖-1 (𝑥 1 , … , 𝑥 𝑘 )𝑝 𝑖+1 (𝑥 1 , … , 𝑥 𝑘 ). (4.12)
See, for example, [29, Lemma 15] for a proof. Let 𝑒 𝑗 (𝐴) ∶= 𝑒 𝑗 (𝜉 𝐴 1 , … , 𝜉 𝐴 𝑘 ) and 𝑝 𝑗 (𝐴) ∶= 𝑝 𝑗 (𝜉 𝐴 1 , … , 𝜉 𝐴 𝑘 ). Similarly define 𝑒 𝑗 (𝐵) and 𝑝 𝑗 (𝐵). By (4.11), we then have

(𝜉 𝐴 1 -𝜉 𝐵 1 ) ⋯ (𝜉 𝐴 1 -𝜉 𝐵 𝑘 ) = 𝑘 ∑ 𝑗=0 (-1) 𝑗 𝑒 𝑗 (𝐵)𝜉 𝑘-𝑗 𝐴 1 (4.13) and 0 = (𝜉 𝐴 1 -𝜉 𝐴 1 ) ⋯ (𝜉 𝐴 1 -𝜉 𝐴 𝑘 ) = 𝑘 ∑ 𝑗=0 (-1) 𝑗 𝑒 𝑗 (𝐴)𝜉 𝑘-𝑗 𝐴 1 . (4.14)
Subtracting (4.14) from (4.13) and using that |𝜉 𝐴 1 -𝜉 𝐵 𝑗 | ⩾ 𝑞𝜅 for any 𝑗 ≠ 1 (which follows from (4.9)) shows that Applying this conclusion to (4.15) then yields that (𝑞𝜅) 𝑘-1 |𝜉 𝐴 1 -𝜉 𝐵 1 | ⩽ 𝛿. But this contradicts the fact that 𝑑(𝐴 1 , 𝐵 1 ) > (𝑞𝜅) -(𝑘-1) 𝛿. Therefore, we must have #𝑆( K1 , … , K𝑘 , □) ⩽ (𝑞𝜅) -𝑘(𝑘-1) which completes the proof of the lemma. □

(𝑞𝜅) 𝑘-1 |𝜉 𝐴 1 -𝜉 𝐵 1 | ⩽ | 𝑘 ∑ 𝑗=0 (-1) 𝑗 (𝑒 𝑗 (𝐵) -𝑒 𝑗 (𝐴))𝜉 𝑘-𝑗 𝐴 1 | ⩽ max 𝑗 |𝑒 𝑗 (𝐵) -𝑒 𝑗 (𝐴)|. ( 4

5

PROOF OF THEOREM 1.1 AND COROLLARY 1.2

Dyadic pigeonholing

It is more convenient to bound

𝐷 𝑝 (𝛿) ∶= sup 𝛿 0 ∈𝑞 -ℕ ∩[𝛿,1] 𝔇 𝑝 (𝛿 0 ) (5.1)
instead of 𝔇 𝑝 (𝛿) as 𝐷 𝑝 (𝛿) is defined for all real 𝛿 ∈ (0, 1] (rather than just for 𝛿 ∈ 𝑞 -ℕ ) and is monotonic, that is, 𝐷 𝑝 (𝛿 𝐿 ) ⩽ 𝐷 𝑝 (𝛿 𝑆 ) if 𝛿 𝐿 ⩾ 𝛿 𝑆 .

Proposition 5.1. For even integers 𝑝 > 2𝑘, there exists a constant 𝐶 > 0, depending only on 𝑘 and 𝑝, such that for every 0 < 𝜀 < 1, we have

𝐷 𝑝 (𝛿) 𝑝 ⩽ 𝐶(log 𝛿 -1 ) 3𝑝 [ 𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 + 𝑞 𝑝 2 + 𝑘 2 +7𝑘-4 2 𝛿 -(𝑘 2 +4𝑘-2)𝜀 𝛿 -1 𝑘 ( 𝑝 2 + 𝑘(𝑘-3) 2 ) 𝐷 𝑝-2𝑘 (𝛿 1-1 𝑘 ) 𝑝-2𝑘 ] (5.2)
for all 0 < 𝛿 < 1.

Proof. To bound 𝐷 𝑝 (𝛿) 𝑝 , suppose 0 < 𝛿 < 1 and 𝛿 0 ∈ 𝑞 ℤ with 𝛿 0 ∈ [𝛿, 1]. We need to bound 𝔇 𝑝 (𝛿 0 ) 𝑝 by decoupling down to frequency scale 𝛿 0 .

Let 𝑓 be a Schwartz function on ℚ 𝑘 𝑞 with Fourier support in

⋃ 𝐾∈𝑃 𝛿 0 𝜃 𝐾 . Then 𝑓 = ∑ 𝐾∈𝑃 𝛿 0 𝑓 𝐾
where f𝐾 ∶= f1 𝐾×ℚ 𝑘-1

𝑞

. We want to prove the existence of 𝐶 > 0 so that for any 0 < 𝜀 < 1,

∫ ℚ 𝑘 𝑞 |𝑓| 𝑝 ⩽ 𝐶(log 𝛿 -1 ) 3𝑝 × [ 𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 + 𝑞 𝑝 2 + 𝑘 2 +7𝑘-4 2 𝛿 -(𝑘 2 +4𝑘-2)𝜀 𝛿 -1 𝑘 ( 𝑝 2 + 𝑘(𝑘-3) 2 ) 𝐷 𝑝-2𝑘 (𝛿 1-1 𝑘 ) 𝑝-2𝑘 ] ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 .
(5.3) In fact, we will prove that for any translate 𝑄 of 𝐵 𝛿 -𝑘 0 ∶= {𝑥 ∈ ℚ 𝑘 𝑞 ∶ |𝑥| ⩽ 𝛿 -𝑘 0 }, we have

∫ 𝑄 |𝑓| 𝑝 ⩽ 𝐶(log 𝛿 -1 ) 3𝑝 × [ 𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 + 𝑞 𝑝 2 + 𝑘 2 +7𝑘-4 2 𝛿 -(𝑘 2 +4𝑘-2)𝜀 𝛿 -1 𝑘 ( 𝑝 2 + 𝑘(𝑘-3) 2 ) 𝐷 𝑝-2𝑘 (𝛿 1-1 𝑘 ) 𝑝-2𝑘 ] ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (𝑄) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 .
(5.4) The estimate (5.3) then follows by summing over all such 𝑄's that tile ℚ 𝑘 𝑞 , and applying Minkowski's inequality to bring an 𝓁 𝑝∕2 norm over 𝑄 on the right-hand side into the sum over 𝐾 ∈ 𝑃 𝛿 0 .

Thus, we now turn to the proof of (5.4)

( 𝑓1 𝑄 ) 𝐾 = f1 𝑄 1 𝐾×ℚ 𝑘-1 𝑞 = ( f * 1𝑄 )1 𝐾×ℚ 𝑘-1 𝑞 = ( f1 𝐾×ℚ 𝑘-1 𝑞 ) * 1𝑄 = f𝐾 * 1𝑄 .
As a result, to prove (5.4), it suffices to prove (5.3) under the additional assumption that 𝑓 is supported on 𝑄. As 𝑓 is an arbitrary Schwartz function with Fourier support in ⋃ where here 𝑓 𝐾 ∶ ℚ 𝑘 𝑞 → ℂ, |𝑓 𝐾 | is the absolute value of 𝑓 𝐾 , and the last characteristic function is meant to be the indicator function of the set {𝑥 ∈ ℚ 𝑘 𝑞 ∶ 𝐻∕2 < |𝑓 𝐾 (𝑥)| ⩽ 𝐻}. As 𝑓 𝐾 is supported on 𝐵 𝛿 -𝑘 0 , so is 𝑓 (𝐻) 𝐾 . By Lemma 2.2, as 𝑓 𝐾 is Fourier supported in 𝜃 𝐾 , we then have

𝑓 (𝐻) 𝐾 = ( ∑ 𝑇∈𝕋(𝐾) 𝑓 𝐾 1 𝑇 ) 1 𝐻∕2<|𝑓 𝐾 |⩽𝐻 = ∑ 𝑇∈𝕋(𝐾) (𝑓 𝐾 1 𝑇 )1 𝐻∕2<|𝑓 𝐾 1 𝑇 |⩽𝐻 (5.5)
where the last equality is because |𝑓 𝐾 1 𝑇 | constant on every 𝑇 ∈ 𝕋(𝐾). Again by Lemma 2.2, note that 𝑓 (𝐻) 𝐾 is Fourier supported in 𝜃 𝐾 . Using the terminology of Lemma 2.2, the nonzero wavepackets that make up 𝑓 (𝐻) 𝐾 are all of height ∼ 𝐻. . This shows

Then ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ 𝑓 - ∑ 𝐾∈𝑃 𝛿 0 ∑ 𝐻∈2 ℤ 𝐻 * ∩(𝛿 1+ 𝑘(𝑘-1) 2𝑝 0 𝐻 * ,𝐻 * ] 𝑓 (𝐻) 𝐾 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖𝐿 ∞ (ℚ 𝑘 𝑞 ) ⩽ ∑ 𝐾∈𝑃 𝛿 0 ‖ ‖ ‖ ‖ ‖ ‖ 𝑓 𝐾 1 |𝑓 𝐾 |⩽𝛿 1+ 𝑘(𝑘-1) 2𝑝 0 𝐻 * ‖ ‖ ‖ ‖ ‖ ‖𝐿 ∞ (ℚ 𝑘 𝑞 ) ⩽ 𝛿 -1 0 (𝛿 1+ 𝑘(𝑘-
‖𝑓‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽ ∑ 𝐻∈2 ℤ 𝐻 * ∩(𝛿 1+ 𝑘(𝑘-1) 2𝑝 0 𝐻 * ,𝐻 * ] ‖ ‖ ‖ ∑ 𝐾∈𝑃 𝛿 0 𝑓 (𝐻) 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) + ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 1∕2 .
Next we dyadically pigeonhole so that each relevant 𝑓 (𝐻) 𝐾 is made up of about the same number of wavepackets. Let now 𝜈 = 𝑞 ⌊log 𝑞 𝛿 1∕𝑘 0 ⌋ ⩽ 𝛿 Finally, we dyadically pigeonhole so that given a 𝐾, the parent interval 𝐽 of length 𝜈 has about the same number of children 𝐾 ′ of length 𝛿 0 such that 𝑓 (𝐻,𝛼) 𝐾 ′ ≠ 0. To be more precise, fix a 𝐾 and let 𝐽 be the unique parent interval of length 𝜈 containing 𝐾. This parent 𝐽 contains 𝜈∕𝛿 0 many intervals 𝐾 ′ of length 𝛿 0 and hence 𝐽 has at most 𝜈∕𝛿 0 many children 𝐾 ′ such that 𝑓 (

∑ 𝛼∈2 ℕ ∩[1,𝛿 -𝑘(𝑘-1)∕2 0 ] ∑ 𝛽∈2 ℕ ∩[1,𝜈∕𝛿 0 ] ‖ ∑ 𝐾∈𝑃 𝛿 0 𝑓 (𝐻,𝛼,𝛽) 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) + ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 1∕2 .
Fix now 𝜀 > 0. For each of the ≲ (log 𝛿 -1 0 ) 3

⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 0 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 = ⎛ ⎜ ⎜ ⎝ ∑ 𝐽∈𝑃 𝜈 ∑ K∈𝑃 𝛿 0 (𝐽) ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 = ⎛ ⎜ ⎜ ⎝ ∑ 𝐽∈𝑃 (𝐻,𝛼,𝛽) 𝜈 ∑ K∈𝑃 𝛿 0 (𝐽) ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 ∼ (𝑁𝛽𝐻) 𝑘
(5.12)

as there are 𝑁 such 𝐽 for which g 𝐽 ≠ 0 and by how g is defined, each of these 𝐽's that contribute has ∼ 𝛽 children K such that g K = 𝑓 (𝐻,𝛼) K ≠ 0. We can finish this estimate once again by using that g K has height ∼ 𝐻. Third,

max 𝐽∈𝑃 𝜈 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 (𝐽) ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ (𝑝-2𝑘)∕2 = max 𝐽∈𝑃 𝜈 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 (𝐽) ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (𝐵 𝛿 -𝑘 0 ) ⎞ ⎟ ⎟ ⎠ (𝑝-2𝑘)∕2
∼ 𝑝,𝑘 𝛽 (𝑝-2𝑘)∕2 𝐻 𝑝-2𝑘 𝛼𝛿 -𝑘(𝑘+1)∕2 0

(5.13) as by how g is defined, the ∑ 𝐾∈𝑃 𝛿 0 (𝐽) has ∼ 𝛽 terms and each term is made up of ∼ 𝛼 wavepackets of height ∼ 𝐻. Note here we made use that each 𝑇 ∈ 𝕋(𝐾) has volume 𝛿 -𝑘(𝑘+1)∕2 0 and g 𝐾 is supported on 𝐵 𝛿 -𝑘 0 . Finally, a similar computation gives that

⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 = ⎛ ⎜ ⎜ ⎝ ∑ 𝐽∈𝑃 𝜈 ∑ 𝐾∈𝑃 𝛿 0 (𝐽) ‖g 𝐾 ‖ 2 𝐿 𝑝 (𝐵 𝛿 -𝑘 0 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 ∼ 𝑝,𝑘 (𝑁𝛽) 𝑝∕2 𝐻 𝑝 𝛼𝛿 -𝑘(𝑘+1)∕2 0 . ( 5
‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎛ ⎜ ⎜ ⎝ ∑ K∈𝑃 𝛿 0 ‖g K ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑘 max 𝐽∈𝑃 𝜈 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 (𝐽) ‖g 𝐾 ‖ 2 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ (𝑝-2𝑘)∕2 ∼ 𝑝,𝑘 𝑁 -(𝑝-2𝑘)∕2 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 .
Using this with Lemma 4.2 where g is as given in (5.9), then shows that

𝜅 ) 𝑝 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 + 𝐶𝑞 -𝑘(𝑘-1) 𝜅 -(𝑘 2 +4𝑘-2) 𝜈 -𝑘(𝑘-1) 2 𝑁 𝑝-2𝑘 2 𝔇 𝑝-2𝑘 ( 𝛿 0 𝜈 ) 𝑝-2𝑘⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 . Note 𝛿 0 𝜅 ⩾ 𝛿 0 𝛿 𝜀 0 = 𝛿 1-𝜀 0 ⩾ 𝛿 1-𝜀 , so 𝔇 𝑝 ( 𝛿 0 𝜅 ) ⩽ 𝐷 𝑝 ( 𝛿 0 𝜅 ) ⩽ 𝐷 𝑝 (𝛿 1-𝜀 )
where in the second inequality, we have used monotonicity. Similarly,

𝛿 0 𝜈 ⩾ 𝛿 0 𝛿 1∕𝑘 0 = 𝛿 1-1 𝑘 0 ⩾ 𝛿 1-1 𝑘 , so 𝔇 𝑝-2𝑘 ( 𝛿 0 𝜈 ) ⩽ 𝐷 𝑝-2𝑘 (𝛿 1-1 𝑘 ). As a result, ∫ ℚ 𝑘 𝑞 |g| 𝑝 ⩽ 𝐶𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 + 𝐶𝑞 -𝑘(𝑘-1) 𝜅 -(𝑘 2 +4𝑘-2) 𝜈 -𝑘(𝑘-1) 2 𝑁 𝑝-2𝑘 2 𝐷 𝑝-2𝑘 (𝛿 1-1 𝑘 ) 𝑝-2𝑘 ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 . Now use 𝑁 ⩽ 𝜈 -1 and ‖g 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) = ‖𝑓 (𝐻,𝛼,𝛽) 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽ ‖𝑓 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) . Thus, ∫ ℚ 𝑘 𝑞 |𝑓| 𝑝 ⩽ 𝐶(log 𝛿 -1 ) 3𝑝 × [ 𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 + 𝑞 -𝑘(𝑘-1) 𝜅 -(𝑘 2 +4𝑘-2) 𝜈 -( 𝑝 2 + 𝑘(𝑘-3) 2 ) 𝐷 𝑝-2𝑘 (𝛿 1-1 𝑘 ) 𝑝-2𝑘 ] ⎛ ⎜ ⎜ ⎝ ∑ 𝐾∈𝑃 𝛿 0 ‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⎞ ⎟ ⎟ ⎠ 𝑝∕2 .
But 𝜈 -1 ⩽ 𝑞𝛿 -1∕𝑘 0 ⩽ 𝑞𝛿 -1∕𝑘 and 𝜅 -1 ⩽ 𝑞𝛿 -𝜀 0 ⩽ 𝑞𝛿 -𝜀 . This completes the proof of (5.2). □

Proof of Theorem 1.1

We now finish the proof of Theorem 1.1. It suffices to iterate (5.2) by using an induction on 𝑝 and induction on 𝛿. Applying the definition of 𝐷 𝑝 (𝛿) from (5.1) and the hypothesis of Theorem 1.1 gives that

𝐷 𝑝 0 (𝛿) 𝑝 0 ⩽ 𝐶 1 𝛿 -( 𝑝 0 2 -𝑘(𝑘+1) 2 )-𝑐(𝑝 0 )(1-1 𝑘 ) 𝑝 0 ∕(2𝑘)
for all 𝛿 ∈ (0, 1) and some 𝑐(𝑝 0 ) ⩾ 0 such that the power of 𝛿 -1 is nonnegative. Note that from (1.9), 𝑎(𝑝, 𝑝 0 ) = 𝑎(𝑝 -2𝑘, 𝑝 0 ) + 𝑝 2 + 𝑘

𝑏 ′ (𝑝) = (1 - 1 𝑘 ) -𝑝 2𝑘 [1 + 𝑝 -𝑘(𝑘 + 1) 2𝑘 log(1 - 1 𝑘 ) -1 ] = (1 - 1 𝑘 ) -𝑝 2𝑘 [1 + 𝑝 -𝑘(𝑘 + 1) 2𝑘 (log 𝑘 -log(𝑘 -1))] ⩾ (1 - 1 𝑘 ) -𝑝 2𝑘 [1 + 2 -𝑘(𝑘 + 1) 2𝑘 (log 𝑘 -log(𝑘 -1))] ⩾ (1 - 1 𝑘 ) -𝑝 2𝑘 [1 + 2 -𝑘(𝑘 + 1) 2𝑘 1 𝑘 -1 ],
where we used 𝑝 ⩾ 2 in the first inequality, and used log 𝑘 -log(𝑘 -1) ⩽ 1 𝑘-1 with 2 -𝑘(𝑘 + 1) ⩽ 0 in the second inequality. This gives

𝑏 ′ (𝑝) ⩾ (1 - 1 𝑘 ) -𝑝 2𝑘 [1 + 2 -𝑘 -𝑘 2 2𝑘(𝑘 -1) ] = (1 - 1 𝑘 ) -𝑝 2𝑘 [1 - 𝑘 + 2 2𝑘 ] ⩾ 0 since 𝑘 ⩾ 2, proving that 𝑏(𝑝) is an increasing function of 𝑝 on [2, ∞).
As a result, from 𝑏(𝑝 0 ) ⩾ 0, we see that 𝑏(𝑝) ⩾ 0 for all 𝑝 ⩾ 𝑝 0 , and hence

𝑝 2 - 𝑘(𝑘 + 1) 2 + 𝑐(𝑝 0 )(1 - 1 𝑘 ) 𝑝 2𝑘 ⩾ 0 (5.15)
for all 𝑝 ⩾ 𝑝 0 . Assume for every 0 < 𝜀 < 1 and all 𝛿 ∈ (0, 1) we know

𝐷 𝑝-2𝑘 (𝛿) 𝑝-2𝑘 ⩽ 𝐶 𝑝-2𝑘,𝜀 𝑞 𝑎(𝑝-2𝑘,𝑝 0 ) 𝛿 -( 𝑝-2𝑘 2 -𝑘(𝑘+1) 2 )-𝑐(𝑝 0 )(1-1 𝑘 ) 𝑝-2𝑘 2𝑘 -𝜀
for some 𝑝 ∈ 𝑝 0 + 2𝑘ℕ (this is true for 𝑝 = 𝑝 0 + 2𝑘) and 𝐶 𝑝-2𝑘,𝜀 is allowed to depend on 𝐶 1 . Then (5.2) gives

𝐷 𝑝 (𝛿) 𝑝 ⩽ 𝐶(log 𝛿 -1 ) 3𝑝 [ 𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 +𝐶 𝑝-2𝑘,𝜀 𝑞 𝑎(𝑝,𝑝 0 ) 𝛿 -(𝑘 2 +4𝑘-2)𝜀 𝛿 -1 𝑘 ( 𝑝 2 + 𝑘(𝑘-3) 2 ) 𝛿 -(1-1 𝑘 )( 𝑝-2𝑘 2 -𝑘(𝑘+1) 2 )-𝑐(𝑝 0 )(1-1 𝑘 ) 𝑝-2𝑘 2𝑘 +1 -𝜀 ] = 𝐶(log 𝛿 -1 ) 3𝑝 𝐷 𝑝 (𝛿 1-𝜀 ) 𝑝 + 𝐶𝐶 𝑝-2𝑘,𝜀 𝑞 𝑎(𝑝,𝑝 0 ) 𝛿 -( 𝑝 2 -𝑘(𝑘+1) 2 )-𝑐(𝑝 0 )(1-1 𝑘 ) 𝑝 2𝑘 -(𝑘 2 +4𝑘)𝜀
for all 𝛿, 𝜀 ∈ (0, 1) where 𝐶 here depends only on 𝑘 and 𝑝. Iterating this inequality 𝑀 times with 𝑀 to be chosen later gives that 𝐷 𝑝 (𝛿) 𝑝 ⩽ 𝐶 𝑀 (log 𝛿 -1 ) 3𝑀𝑝 𝐷 𝑝 (𝛿 (1-𝜀) 𝑀 ) 𝑝 + 𝐶𝐶 𝑝-2𝑘,𝜀 𝑞 𝑎(𝑝,𝑝 0 ) 𝛿 -(𝑘 Trivially, we have 𝐷 𝑝 (𝛿 (1-𝜀) 𝑀 ) ⩽ 𝛿 -(1-𝜀) 𝑀 ∕2 . Thus,

𝐷 𝑝 (𝛿) 𝑝 ⩽ 𝐶 𝑀 (log 𝛿 -1 ) 3𝑀𝑝 𝛿 -(1-𝜀) 𝑀 𝑝∕2 + 𝐶𝐶 𝑝-2𝑘,𝜀 𝑞 𝑎(𝑝,𝑝 0 ) 𝛿 -(𝑘 2 +4𝑘)𝜀 𝑀-1 ∑ 𝑗=0 𝐶 𝑗 (log 𝛿 -1 ) 3𝑝𝑗 𝛿 -(1-𝜀) 𝑗 [( 𝑝 2 -𝑘(𝑘+1) 2 )+𝑐(𝑝 0 )(1-1 𝑘 ) 𝑝 2𝑘 ] .
(5.16) By (5.15), the power of 𝛿 -1 in (5.16) is positive and so using that (1 -𝜀) 𝑗 ⩽ 1, the sum can be controlled by 𝑀𝐶 𝑀 (log 𝛿 -1 ) 3𝑀𝑝 𝛿 -( 𝑝 2 -𝑘(𝑘+1) 2 )-𝑐(𝑝 0 )(1-1 𝑘 ) 𝑝 2𝑘 .

Inserting this into (5.16) and choosing 𝑀 be the least integer such that (1 -𝜀) 𝑀 ⩽ 𝜀 (and so 𝑀 = ⌈ ) 𝑘 holds for every Schwartz function g on ℚ 𝑘 𝑞 with Fourier transform supported in ⋃ 𝐾∈𝑃 𝛿 𝜃 𝐾 . We will prove that 𝑆(𝛿) ≲ 𝜀 𝛿 -𝜀 for every 𝜀 > 0, which by Minkowski's inequality is stronger than the assertion 𝔇 2𝑘 (𝛿) ≲ 𝜀 𝛿 -𝜀 .

Let 𝛿 ∈ 𝑞 -ℕ , g be as above, and 𝜅 ∈ 𝑞 ) 𝑘 , which for the purposes below is as good as (A.3). Putting (A.2) and (A.3) back into (A.1), we have 𝑆(𝛿) 2𝑘 ⩽ 2 2𝑘-1 𝑘 2𝑘 𝑆( 𝛿 𝜅 ) 2𝑘 + 2 2𝑘-1 𝜅 -(4𝑘-2) (𝑞𝜅) -𝑘(𝑘-1) .

1 2

 1 𝑘 2 (1-1 𝑘 ) 𝑠∕𝑘

. 15 )

 15 Next we claim that |𝑒 𝑗 (𝐵) -𝑒 𝑗 (𝐴)| ⩽ 𝛿 for all 𝑗 = 1, 2, … , 𝑘. We prove this by induction. As 𝑒 1 = 𝑝 1 , |𝑒 1 (𝐵) -𝑒 1 (𝐴)| ⩽ 𝛿 by the 𝑗 = 1 case of (4.10). Now assume that for some 𝐽 = 1, 2, … , 𝑘 -1 we had |𝑒 𝑗 (𝐵) -𝑒 𝑗 (𝐴)| ⩽ 𝛿 for all 𝑗 = 1, 2, … , 𝐽. Then by (4.12),|(𝐽 + 1)𝑒 𝐽+1 (𝐵) -(𝐽 + 1)𝑒 𝐽+1 (𝐴)| = | 𝐽 ∑ 𝑖=0 (-1) 𝑖 (𝑒 𝐽-𝑖 (𝐵)𝑝 𝑖+1 (𝐵) -𝑒 𝐽-𝑖 (𝐴)𝑝 𝑖+1 (𝐴))| ⩽ max 0⩽𝑖⩽𝐽 |𝑒 𝐽-𝑖 (𝐵)𝑝 𝑖+1 (𝐵) -𝑒 𝐽-𝑖 (𝐴)𝑝 𝑖+1 (𝐴)|.20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Observe that |𝑒 𝐽-𝑖 (𝐵)𝑝 𝑖+1 (𝐵) -𝑒 𝐽-𝑖 (𝐴)𝑝 𝑖+1 (𝐴)| = |𝑒 𝐽-𝑖 (𝐵)(𝑝 𝑖+1 (𝐵) -𝑝 𝑖+1 (𝐴)) + 𝑝 𝑖+1 (𝐴)(𝑒 𝐽-𝑖 (𝐵) -𝑒 𝐽-𝑖 (𝐴))| ⩽ max(|𝑝 𝑖+1 (𝐵) -𝑝 𝑖+1 (𝐴)|, |𝑒 𝐽-𝑖 (𝐵) -𝑒 𝐽-𝑖 (𝐴)|) ⩽ 𝛿 by the inductive hypothesis and (4.10). As 𝑞 is a prime > 𝑘, it follows that |𝑒 𝐽+1 (𝐵) -𝑒 𝐽+1 (𝐴)| ⩽ 𝛿.

.

  Thus from now on, we assume additionally that 𝑓 and all the 𝑓 𝐾 are supported on 𝐵 𝛿 -𝑘 0 and prove (5.3). We first dyadically pigeonhole 𝑓 by wavepacket height. Write 𝐻 * = max 𝐾∈𝑃 𝛿 0 ‖𝑓 𝐾 ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) . For 𝐾 ∈ 𝑃 𝛿 0 and 𝐻 ∈ 2 ℤ 𝐻 * ∩ (𝛿 1+ 𝑘(𝑘-1) 2𝑝 0 𝐻 * , 𝐻 * ], let 𝑓 (𝐻) 𝐾 = 𝑓 𝐾 1 𝐻∕2<|𝑓 𝐾 |⩽𝐻

1∕𝑘 0 ..,

 0 From (5.5), 𝑓(𝐻) 𝐾 is Fourier supported in 𝜃 𝐾 and supported in 𝐵 𝛿 -𝑘 0 As a 𝑇 ∈ 𝕋(𝐾) is either completely contained in or completely disjoint from 𝐵 𝛿 1 𝑇 )1 𝐻∕2<|𝑓 𝐾 1 𝑇 |⩽𝐻 .(5.6)Furthermore, the 𝑇 ∈ 𝕋(𝐾) that are contained in 𝐵 𝛿 -𝑘 0 perfectly partition 𝐵 𝛿 -𝑘 0 into 𝛿 -𝑘(𝑘-1)∕2 0 many translates of 𝑇 0,𝑘 . Thus, (5.6) has at most 𝛿 -𝑘(𝑘-1)∕2 0 many nonzero terms. Therefore for 𝛼 ∈ 2 ℕ ∩ [1of nonzero terms in (5.6) (that is, the number of nonzero wavepackets in 𝑓 (𝐻) 𝐾 ) is in (𝛼∕2, 𝛼], and 0 otherwise. Thus, now we have that that is supported in 𝐵 𝛿 -𝑘 0 Fourier supported in 𝜃 𝐾 , and has ∼ 𝛼 many nonzero wavepackets of height ∼ 𝐻.

log 𝜀 - 1 log( 1 -

 11 𝜀) -1 ⌉) then shows that 𝐷 𝑝 (𝛿) ≲ 𝑝,𝜀,𝐶 1 𝑞 𝑎(𝑝,𝑝 0 )∕𝑝 (log 𝛿 -1 ) 𝑘 2 +4𝑘)𝜀 𝑝 for all 𝛿, 𝜀 ∈ (0, 1). As (log 𝛿 -1 ) 3𝑀 ≲ 𝜀 𝛿 -𝜀 , by redefining 𝜀 we have 𝐷 𝑝 (𝛿) ≲ 𝑝,𝜀,𝐶 1 𝑞 𝑎(𝑝,𝑝 0 )∕𝑝 𝛿 This question was first posed to the third and sixth author by Shaoming Guo when the third author was visiting the Department of Mathematics at the Chinese University of Hong Kong in July 2019. This question was posed again by Shaoming Guo during a problem session at the Arithmetic (and) Harmonic Analysis workshop held (virtually) at the Mittag-Leffler Institute in early June 2021 and this current collaboration arose from that particular workshop.Kevin Hughes is supported by the Additional Funding Programme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical Research. Zane Kun Li is supported by NSF Grant DMS-1902763. Akshat Mudgal is supported by Ben Green's Simons Investigator Grant, ID 376201; Olivier Robert is supported by the joint FWF-ANR Project Arithrand: FWF: I 4945-N and ANR-20-CE91-0006; and Po-Lam Yung is supported by a Future Fellowship FT20010039 from the Australian Research Council. Zane Kun Li would also like to thank the National Center for Theoretical Sciences (NCTS) in Taipei, Taiwan for their kind hospitality during his visit, where part of this work was written. The authors also acknowledge kind support from the American Institute of Mathematics through the Fourier restriction research community.J O U R N A L I N F O R M AT I O NMathematika is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.

20417942, 2024, 1 ,

 1 Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseAPPENDIX: PROOF OF 𝕯 𝟐𝒌 (𝜹) ≲ 𝜺 𝜹 -𝜺Fix 𝑘 ∈ ℕ and a prime 𝑞 > 𝑘. For 𝛿 ∈ 𝑞 -ℕ , let 𝑆(𝛿) be the smallest constant such that the reverse square function estimate

  1, … , 𝑞 -1} for every 𝑗 ⩾ 𝑘 } , and the 𝑞-adic norm on ℚ 𝑞 satisfies | ∑ ∞ 𝑗=𝑘 𝑎 𝑗 𝑞 𝑗 | = 𝑞 -𝑘 if 𝑎 𝑘 ≠ 0. Strictly speaking we should be writing | ⋅ | 𝑞 instead of | ⋅ |, but we omit this dependence as 𝑞 is fixed. The 𝑞-adic norm on ℚ 𝑞 induces a norm on ℚ 𝑘 𝑞 , which we denote also by | ⋅ | by abuse of notation, via |(𝜉 1 , … , 𝜉 𝑘 )| ∶= max 1⩽𝑖⩽𝑘 |𝜉 𝑖 |. Of particular importance is the ultrametric inequality: |𝜉 + 𝜂| ⩽ max{|𝜉|, |𝜂|} with equality if |𝜉| ≠ |𝜂|. An interval in ℚ 𝑞

  𝐾∈𝑃 𝛿 𝜃 𝐾 . Note that 𝑓 = ∑ 𝐾∈𝑃 𝛿 𝑓 𝐾 . Bourgain, Demeter, and Guth

20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License holds for every Schwartz function 𝑓 on ℚ 𝑘 𝑞 with its Fourier transform f supported on ⋃

  20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

)

  20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewhere 𝑥 1 , … , 𝑥 𝑘 are distinct mod 𝑝 and 𝑦 1 , … , 𝑦 𝑘 are distinct mod 𝑝 and since we are considering 𝐽 𝑠,𝑘 (𝑋, 𝑝, 0), we have that 𝑥 𝑘+1 , … , 𝑥 𝑠 , 𝑦 𝑘+1 , … , 𝑦 𝑠 ≡ 0 (mod 𝑝). Each choice of 𝑥 1 , … , 𝑥 𝑘 , 𝑦 1 , … , 𝑦 𝑘 gives ⩽ 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝) many solutions to (𝑥 𝑘+1 , … , 𝑥 𝑠 , 𝑦 𝑘+1 , … , 𝑦 𝑠 ). To see this, write the count for (3.2) as an integral and use the triangle inequality; the basic idea being that shifts of the Vinogradov system can only give fewer solutions.Next, fixing one of the at most 𝐽 𝑠-𝑘,𝑘 (𝑋∕𝑝) many tuples (𝑥 𝑘+1 , … , 𝑥 𝑠 , 𝑦 𝑘+1 , … , 𝑦 𝑠 ), how many valid 𝑥 1 , … , 𝑥 𝑘 , 𝑦 1 , … , 𝑦 𝑘 are there? As requiring 𝑦 1 , … , 𝑦 𝑘 to be distinct mod 𝑝 is a rather mild condition, there are ⩽ 𝑋 𝑘 such (𝑦 1 , … , 𝑦 𝑘 ). Any valid (𝑥 1 , … , 𝑥 𝑘 ) ∈ ([1, 𝑋] ∩ ℕ) 𝑘 must satisfy

  Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons LicenseProof. This proof is standard and follows from a change of variables that can be found, for example, in[8, section 11.2]. □ Our main lemma in proving Theorem 1.1 is the following: Lemma 4.2. Let 𝑝 ∈ 2𝑘 + 2ℕ, 𝛿 ∈ 𝑞 -ℕ and 𝜅 ∈ 𝑞 -ℕ ∩ [𝛿, 1). Let 𝜈 = 𝑞 ⌊log 𝑞 𝛿 1∕𝑘 ⌋ ∈ 𝑞 -ℕ so that 𝜈 ⩽ 𝛿 1∕𝑘 . If g is a Schwartz function with Fourier support in ⋃ 𝐾∈𝑃 𝛿 𝜃 𝐾 , then we have

	20417942, 2024, 1, ∫ ℚ 𝑘 𝑞 |g| 𝑝 ⩽ 𝐶𝔇 𝑝 ( 𝛿 𝜅 ) 𝑝 ( ∑ 𝐾∈𝑃 𝛿 ‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )
							Vinogradov
	system (1.1).
	Lemma 4.1 (Affine rescaling). Let g be a Schwartz function on ℚ 𝑘 𝑞 Fourier supported in Then for any interval 𝐼 ⊂ ℤ 𝑞 of length 𝜅 ⩾ 𝛿, we have	⋃	𝐾∈𝑃 𝛿 𝜃 𝐾 .
	‖g 𝐼 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽ 𝔇 𝑝	(	𝛿 𝜅	) ⎛ ⎜ ⎜ ⎝ 𝐾∈𝑃 𝛿 (𝐼) ∑	‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )	1∕2 ⎞ ⎟ ⎠ ⎟	.

2

  20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License with 𝑎 𝐾 = 𝑏 𝐾 = ‖g 𝐾 ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) and 𝑐 𝐾 = ‖g 𝐾 ‖ 𝐿 𝑝-2𝑘 (ℚ 𝑘 𝑞 ) , we get

	( 𝐾∈𝑃 𝛿 ∑	‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )

  𝑥 be the set of all intervals 𝐼 ′ ∈ 𝑃 𝜅 such that |g 𝐼 ′ (𝑥)| ⩾ 𝜅 max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)|. Suppose first  𝑥 contains at least 𝑘 20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License (disjoint) intervals, say 𝐼 ′ 1 , … , 𝐼 ′ 𝑘 (all dependent on 𝑥) of length 𝜅 and |g 𝐼 ′ 1 (𝑥)| = max 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)|: in this case we have |g

First, we have the pointwise bound |g(𝑥)| ⩽ ∑ 𝐼∈𝑃 𝜅 |g 𝐼 (𝑥)|. At every point 𝑥 ∈ ℚ 𝑘 𝑞 , let 

  …,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 𝐼 𝑘 | 2 |g| 𝑝-2𝑘 . (4.3) This inequality (4.3) is the analogue of Step 1 in Subsection 3.1. The requirement that we analyze solutions to the Vinogradov system with 𝑥 1 , … , 𝑥 𝑠 and 𝑦 1 , … , 𝑦 𝑠 being distinct mod 𝑝 corresponds to the requirement that we analyze ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 |g| 𝑝-2𝑘 with 𝑑(𝐼 𝑖 , 𝐼 𝑗 ) > 𝜅 for all 1 ⩽ 𝑖 ≠ 𝑗 ⩽ 𝑘

	𝑞 ∫ ℚ 𝑘	|g| 𝑝
	≲ 𝑝 𝔇 𝑝 (	𝛿 𝜅	) 𝑝	(	∑ 𝐾∈𝑃 𝛿	‖g 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )
	∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g with 𝜅 = 1∕𝑞.
	Next, we mimic Step 2 in Subsection 3.2. Recalling our definition of 𝑁 in the statement of
	Lemma 4.2, Hölder's inequality gives
							|g| 𝑝-2𝑘 ⩽ 𝑁 𝑝-2𝑘-1	∑	|g 𝐽 | 𝑝-2𝑘 .
							𝐽∈𝑃 𝜈

Applying this in the second term in (4.3), we get 20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

  20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on[12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Lemma 4.4. Let 𝑝 ∈ 2𝑘 + 2ℕ, 𝛿 ∈ 𝑞 -ℕ , 𝜅 ∈ 𝑞 -ℕ ∩ [𝛿, 1), and 𝜈 = 𝑞 ⌊log 𝑞 𝛿 1∕𝑘 ⌋ ∈ 𝑞 -ℕ so that 𝜈 ⩽ 𝛿 1∕𝑘 . Suppose that 𝐼 1 , … , 𝐼 𝑘 ∈ 𝑃 𝜅 with 𝑑(𝐼 𝑖 , 𝐼 𝑗 ) > 𝜅 for all 𝑖 ≠ 𝑗. Let □ be a cube of side length 𝜈 𝑘 and K1 ∈ 𝑃 𝛿 (𝐼 1 ), … , K𝑘 ∈ 𝑃 𝛿 (𝐼 𝑘 ). Define 𝑆( K1 , … , K𝑘 , □) be the set of all ordered 𝑘-tuples (𝐾 1 , … , 𝐾 𝑘 ) with 𝐾 𝑖 ∈ 𝑃 𝛿 (𝐼 𝑖 ) such that 0 ∈ supp( ĝ𝐾 1 * ⋯ * ĝ𝐾 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 * ( |g 𝐽 | 𝑝-2𝑘 1 □ )).

  where here we recall the definition of 𝜏 𝐾 in(1.4). Each 𝜏 𝐴 𝑖 , 𝜏 𝐵 𝑖 , and 𝜏 K𝑖 are cubes in ℚ 𝑘 𝑞 of side length 𝛿 and □ is a cube in ℚ 𝑘 𝑞 of side length 𝜈 𝑘 ⩽ 𝛿. Thus by the ultrametric inequality, both ∑ 𝑘 𝑖=1 𝜏 𝐴 𝑖 -, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Therefore, (after another application of the ultrametric inequality) there exists 𝜉 𝐴 𝑖 ∈ 𝐴 𝑖 and 𝜉 𝐵 𝑖 ∈ 𝐵 𝑖 such that Next for 𝑗 = 1, 2, … , 𝑘, define the elementary symmetric polynomials 𝑒 𝑗 (𝑥 1 , … , 𝑥 𝑘 ) ∶= ∑ 1⩽𝑖 1 <⋯<𝑖 𝑗 ⩽𝑘 𝑥 𝑖 1 ⋯ 𝑥 𝑖 𝑗 . Additionally, let 𝑒 0 (𝑥 1 , … , 𝑥 𝑘 ) ∶= 1. Then we have the two identities:

				| | | | | | ∑ 𝑘 𝑖=1	𝜉 𝐴 𝑖 𝑗	-	𝑘 ∑ 𝑖=1	𝜉 𝐵 𝑖 𝑗	| | | | | |	⩽ 𝛿	(4.10)
	for 𝑗 = 1, 2, … , 𝑘.
	We now use the Newton-Girard identities to derive a contradiction. For 𝑗 = 1, 2, … , 𝑘, define the power sums 𝑝 𝑗 (𝑥 1 , … , 𝑥 𝑘 ) ∶= 𝑥 𝑗 1 + ⋯ + 𝑥 𝑗 𝑘 .
	By (4.7) and (4.8), we have that
	0 ∈	( 𝑘 ∑	𝜏 𝐴 𝑖 -	𝑘 ∑	𝜏 K𝑖 + □ )	∩	( 𝑘 ∑	𝜏 𝐵 𝑖 -	𝑘 ∑	𝜏 K𝑖 + □ )
		𝑖=1		𝑖=1							𝑖=1	𝑖=1
	∑ 𝑘 𝑖=1 𝜏 K𝑖 + □ and	∑ 𝑘 𝑖=1 𝜏 𝐵 𝑖 -	∑ 𝑘 𝑖=1 𝜏 K𝑖 + □ are cubes in ℚ 𝑘 𝑞 of side length 𝛿. Further-
	more, by the ultrametric inequality, as two cubes of side length 𝛿 are either completely disjoint or
	exactly the same, we must have
		𝑘 ∑	𝜏 𝐴 𝑖 -	𝑘 ∑		𝜏 K𝑖 + □ =	𝑘 ∑	𝜏 𝐵 𝑖 -	𝑘 ∑	𝜏 K𝑖 + □
		𝑖=1		𝑖=1							𝑖=1	𝑖=1
	and hence
			𝑘 ∑	𝜏 𝐴 𝑖 -
				𝑖=1

𝑖 = 1, 2, … , 𝑘, we have 𝐴 𝑖 , 𝐵 𝑖 ⊂ 𝐼 𝑖 and 𝑑(𝐼 𝑖 , 𝐼 𝑗 ) > 𝜅 for all 𝑖 ≠ 𝑗, this implies 𝑑(𝐴 𝑖 , 𝐴 𝑗 ) ⩾ 𝑞𝜅, 𝑑(𝐵 𝑖 , 𝐵 𝑗 ) ⩾ 𝑞𝜅, 𝑑(𝐴 𝑖 , 𝐵 𝑗 ) ⩾ 𝑞𝜅 whenever 𝑗 ≠ 𝑖 (4.9) (thus the only distances we do not have any control over are the ones of the form 𝑑(𝐴 𝑖 , 𝐵 𝑖 ), 𝑖 ≠ 1). 𝑘 ∑ 𝑖=1 𝜏 𝐵 𝑖 = 𝐵(0, 𝛿).

20417942

  20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License COOK et al.

  . Note that for any translate 𝑄 of 𝐵 𝛿 -𝑘 𝑃 𝛿 0 . Next, we have (𝑓1 𝑄 ) 𝐾 = 𝑓 𝐾 1 𝑄 ; indeed

	supported in 𝐵 𝛿 𝑘 0 for all 𝐾 ∈	. Therefore, 𝑓1 𝑄 is still Fourier supported in	⋃	𝐾∈𝑃 𝛿 0	0 𝜃 𝐾 because 𝜃 𝐾 + 𝐵 𝛿 𝑘 , we have that 1𝑄 is = 𝜃 𝐾 0

  Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on[12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License so as 𝑓 and 𝑓(𝐻) 𝐾 are supported on 𝐵 𝛿 -𝑘

				20417942, 2024, 1, 0 ,
	‖𝑓 -	∑ 𝐾∈𝑃 𝛿 0	∑ 𝐻∈2 ℤ 𝐻 * ∩(𝛿 0 1+ 𝑘(𝑘-1) 2𝑝	𝐻 * ,𝐻 * ]	𝑓 (𝐻) 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽ (𝛿	𝑘(𝑘-1) 2𝑝 0	𝐻 * )|𝐵 𝛿 -𝑘 0	|	1 𝑝 = 𝐻 * 𝛿	-𝑘(𝑘+1) 2𝑝 0
					⩽ max 𝐾∈𝑃 𝛿 0	‖𝑓 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽	⎛ ⎜ ⎜ ⎝ 𝐾∈𝑃 𝛿 0 ∑	‖𝑓 𝐾 ‖ 2 𝐿 𝑝 (ℚ 𝑘 𝑞 )	⎠ ⎟ ⎟ ⎞ 1∕2
	where the second inequality follows from writing 𝑓 𝐾 = 𝑓 𝐾 * 1 ⋀ 𝜃 𝐾 and applying Young's inequality
	‖𝑓 𝐾 ‖ 𝐿 ∞ (ℚ 𝑘 𝑞 ) ⩽ ‖𝑓 𝐾 ‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ‖1 ⋀ 𝜃 𝐾 ‖ 𝐿 𝑝 ′ (ℚ 𝑘 𝑞 )
									1)	𝑘(𝑘-1)
								0	2𝑝		𝐻 * ) = 𝛿	0	2𝑝	𝐻 *

  Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 𝐾 ⊂ 𝐽 and 𝛽 ∈ 2 ℕ ∩ [1, 𝜈∕𝛿 0 ], let 𝐾 if the number of children 𝐾 ′′ of 𝐽 with 𝑓 (𝐻,𝛼) 𝐾 ′′≠ 0 is in (𝛽∕2, 𝛽], that is, if #{𝐾 ′′ ∈ 𝑃 𝛿 0 (𝐽) ∶ 𝑓(𝐻,𝛼) , Fourier supported in 𝜃 𝐾 , has ∼ 𝛼 many nonzero wavepackets of height ∼ 𝐻, and 𝐾's parent 𝐽 has ∼ 𝛽 children each of which also are supported in 𝐵 𝛿 -𝑘 0 , Fourier supported in 𝜃 𝐾 , and have ∼ 𝛼 many nonzero wavepackets of height ∼ 𝐻.

			20417942, 2024, 1, 𝑓 (𝐻,𝛼,𝛽) 𝐾 ∶= 𝑓 (𝐻,𝛼)
	0} ∈ (𝛽∕2, 𝛽], and 0 otherwise. Thus, we now have	𝐾 ′′	≠
		𝑓 (𝐻,𝛼) 𝐾	=	∑	𝑓 𝐾 (𝐻,𝛼,𝛽)	(5.8)
					𝛽∈2 ℕ ∩[1,𝜈∕𝛿 0 ]
	and each 𝑓 𝐾 (𝐻,𝛼,𝛽)	is a function that is supported in 𝐵 𝛿 -𝑘 0
	Thus, combining (5.7) and (5.8) gives
		𝑓 (𝐻) 𝐾 =		∑	∑	𝑓 𝐾 (𝐻,𝛼,𝛽)	,
		𝛼∈2 ℕ ∩[1,𝛿	-𝑘(𝑘-1)∕2 0	]	𝛽∈2 ℕ ∩[1,𝜈∕𝛿 0 ]
	which implies
	‖𝑓‖ 𝐿 𝑝 (ℚ 𝑘 𝑞 ) ⩽	∑ 𝐻∈2 ℤ 𝐻 * ∩(𝛿 𝑘(𝑘-1) 1+ 2𝑝 0	𝐻
						𝐻,𝛼) 𝐾 ′	≠ 0. For

* ,𝐻 * ]

  choices of (𝐻, 𝛼, 𝛽), we apply Lemma 4.2 with 𝛿 replaced by 𝛿 0 , to , 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

	With this, we then first compute
							max 𝐾∈𝑃 𝛿 0	‖g 𝐾 ‖ 𝑘 𝐿 ∞ (ℚ 𝑘 𝑞 )	∼ 𝐻 𝑘	(5.11)
	as g 𝐾 = 𝑓	(𝐻,𝛼,𝛽) 𝐾	which has height ∼ 𝐻. Next, we have
					g ∶=	∑	𝑓 𝐾 (𝐻,𝛼,𝛽)	=	∑	∑	𝑓 (𝐻,𝛼) 𝐾	,	(5.9)
						𝐾∈𝑃 𝛿 0				𝐽∈𝑃 𝜈 (𝐻,𝛼,𝛽)	𝐾∈𝑃 𝛿 0 (𝐽)
	where
			𝑃	(𝐻,𝛼,𝛽) 𝜈	= {𝐽 ∈ 𝑃 𝜈 ∶ #{𝐾 ′′ ∈ 𝑃 𝛿 0 (𝐽) ∶ 𝑓 (𝐻,𝛼) 𝐾 ′′	≠ 0} ∈ (𝛽∕2, 𝛽]}
	and 𝜅 ∶= 𝑞 ⌊log 𝑞 𝛿 𝜀 0 ⌋ ⩽ 𝛿 𝜀 0 . Note that this implies
						g 𝐽 = 1 𝑃 (𝐻,𝛼,𝛽) 𝜈	(𝐽)	𝐾∈𝑃 𝛿 0 (𝐽) ∑	𝑓 (𝐻,𝛼) 𝐾	(5.10)
	and g 𝐾 = 𝑓 (𝐻,𝛼) 𝐾 of 𝐽 ∈ 𝑃 𝜈 for which g 𝐽 ≠ 0 as in Lemma 4.2, and so 𝑁 = #𝑃 if 𝐾's parent 𝐽 is contained in 𝑃 (𝐻,𝛼,𝛽) 𝜈 and 0 otherwise. Write 𝑁 for the number (𝐻,𝛼,𝛽) 𝜈 . Note that by assumption the number of nonzero terms in the ∑

𝐾 in (5.10) is ∼ 𝛽.

20417942

  20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

	𝑞 ∫ ℚ 𝑘	|g| 𝑝 ⩽ 𝐶𝔇 𝑝 (	𝛿 0

  Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License and 𝑏(𝑝) is an increasing function of 𝑝 on [2, ∞): indeed, for 𝑝 ⩾ 2, we have

				20417942, 2024, 1,
	0 where	2 +7𝑘-4 2	and 𝑎(𝑝 0 , 𝑝 0 ) = 0. Additionally, (1.10) gives 𝑏(𝑝 0 ) ⩾
		𝑏(𝑝) ∶= (𝑝 -𝑘(𝑘 + 1))(1 -	1 𝑘	) -𝑝 2𝑘 + 2𝑐(𝑝 0 ),

  2

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				20417942, 2024, 1,
	+4𝑘)𝜀	𝑀-1 ∑	𝐶 𝑗 (log 𝛿 -1 ) 3𝑝𝑗 𝛿 -(1-𝜀) 𝑗 [( 𝑝 2 -𝑘(𝑘+1) 2 )+𝑐(𝑝 0 )(1-1 𝑘 )	𝑝 2𝑘 ] .
		𝑗=0

  -ℕ ∩ [𝛿, 1]. The broad/narrow dichotomy given by the pointwise estimate (4.2) implies 20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewhere we used the pointwise inequality∑ 𝐼∈𝑃 𝜅 ( ∑ 𝐾∈𝑃 𝛿 (𝐼) |g 𝐾 | 2 ) 𝑘 ⩽ ( ∑ 𝐾∈𝑃 𝛿 |g 𝐾 | 2 )𝑘 in the last inequality. To proceed further, fix now 𝐼 1 , … , 𝐼 𝑘 ∈ 𝑃 𝜅 with 𝑑(𝐼 𝑖 , 𝐼 𝑗 ) > 𝜅 for all 𝑖 ≠ 𝑗. We expand∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 = ∑ 𝐾 1 … g 𝐾 𝑘 g K1 … g K𝑘 𝐾 1 … g 𝐾 𝑘 g K1 … g K𝑘 = [ ĝ𝐾 1 * ⋯ * ĝ𝐾 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 ](0).For each K1 ∈ 𝑃 𝛿 (𝐼 1 ), … , K𝑘 ∈ 𝑃 𝛿 (𝐼 𝑘 ), we count the number of ordered 𝑘-tuples (𝐾 1 , … , 𝐾 𝑘 ) with 𝐾 𝑖 ∈ 𝑃 𝛿 (𝐼 𝑖 ) for 𝑖 = 1, … , 𝑘 and 0 ∈ supp( ĝ𝐾 1 * ⋯ * ĝ𝐾 𝑘 * ĝ K1 * ⋯ * ĝ K𝑘 ). The proof of Lemma 4.4shows that the number of such ordered 𝑘-tuples is ⩽ (𝑞𝜅) -𝑘(𝑘-1) (in fact, here we only need that ĝ𝐾 𝑗 is supported in the cube 𝜏 𝐾 𝑗 rather than the smaller parallelepiped 𝜃 𝐾 𝑗 ). So, using Cauchy-Schwarz,∑ 𝐾 1 … g 𝐾 𝑘 g K1 … g K𝑘 ⩽ (𝑞𝜅) -𝑘(𝑘-1) ∑ 𝐾 𝑖 ∈𝑃 𝛿 (𝐼 𝑖 ) 𝑖=1,…,𝑘 ∫ ℚ 𝑘 𝑞 |g 𝐾 1 … g 𝐾 𝑘 | 2 . 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 ⩽ (𝑞𝜅) -𝑘(𝑘-1)Alternatively, multilinear restriction estimate and 𝐿 2 orthogonality says that for any ball 𝐵 𝛿 -1 of radius 𝛿 -1 in ℚ 𝑘 𝑞 , one has∫ 𝐵 𝛿 -1 |g 𝐼 1 … g 𝐼 𝑘 | 2 ≲ 𝜅 (𝛿 𝑘-1 ) 𝑘 𝐵 𝛿 -1 |g 𝐼 𝑗 | 2 = |𝐵 𝛿 -1 | -(𝑘-1)Summing over all 𝐵 𝛿 -1 ⊂ ℚ 𝑘 𝑞 and all 𝐼 1 , … , 𝐼 𝑘 ∈ 𝑃 𝜅 , we have ∑ 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 ∫ ℚ 𝑘 𝑞 |g 𝐼 1 … g 𝐼 𝑘 | 2 ≲ 𝜅 ∫ ℚ 𝑘

	𝐾 𝑖 ∈𝑃 𝛿 (𝐼 𝑖 ) 𝑖=1,…,𝑘 g and write ∑ K𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑗=1,…,𝑘 ∫ ℚ 𝑘 𝑞
	∫ ℚ 𝑘 𝑞 g 𝐾 𝑖 ∈𝑃 𝛿 (𝐼 𝑖 ) 𝑖=1,…,𝑘 ∑ K𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑗=1,…,𝑘 g It follows that ∫ ℚ 𝑘 𝑞
			∑										𝑞 ∫ ℚ 𝑘	𝐾∈𝑃 𝛿 ( ∑	|g 𝐾 | 2	) 𝑘	.	(A.3)
																𝑘 ∏ 𝑗=1	∫ 𝐵 𝛿 -1	⎛ ⎜ ⎜ ⎝ 𝐾 𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) ∑	|g 𝐾 𝑗 | 2	⎞ ⎟ ⎠ ⎟ ,
	and as each |g 𝐾 𝑗 | is constant on 𝐵 -1 𝛿 , we have
	∫ ℚ 𝑘 𝑞	|g| 2𝑘 ⩽ 2 2𝑘-1 𝑘 2𝑘 |𝐵 𝛿 -1 | -(𝑘-1) ∏ ∑ 𝐼∈𝑃 𝜅 𝑘 𝑗=1 ∫ 𝐵 𝛿 -1 ∫ ℚ 𝑘 𝑞	|g 𝐼 | 2𝑘 + 2 2𝑘-1 𝜅 -(4𝑘-2) ⎛ ⎜ ⎜ ⎝ ∑ 𝐾 𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) |g 𝐾 𝑗 | 2 ⎞ ⎟ ⎟ ⎠ = ∫ 𝐵 𝛿 -1 𝐼 1 ,…,𝐼 𝑘 ∈𝑃 𝜅 ∑ 𝑘 ∏ 𝑗=1 ⎛ ⎜ ⎜ ⎝ 𝐾 𝑗 ∈𝑃 𝛿 (𝐼 𝑗 ) 𝑞 ∫ ℚ 𝑘 ∑	|g 𝐼 1 … g 𝐼 𝑘 | 2 |g 𝐾 𝑗 | 2 ⎞ ⎟ ⎠ ⎟ .	(A.1)
	𝑑(𝐼 𝑖 ,𝐼 𝑗 )>𝜅 ∀𝑖≠𝑗 𝐾∈𝑃 𝛿 Furthermore, by a rescaling argument similar to that in Lemma 4.1, we have 𝑞 ( ∑ |g 𝐾 | 2
	∑ 𝐼∈𝑃 𝜅	∫ ℚ 𝑘 𝑞	|g 𝐼 | 2𝑘 ⩽ 𝑆(	𝛿 𝜅	) 2𝑘	∑ 𝐼∈𝑃 𝜅	∫ ℚ 𝑘 𝑞	⎛ ⎜ ⎜ ⎝ 𝐾∈𝑃 𝛿 (𝐼) ∑	|g 𝐾 | 2	⎞ ⎟ ⎟ ⎠ 𝑘	⩽ 𝑆	(	𝛿 𝜅	) 2𝑘	𝑞 ∫ ℚ 𝑘	𝐾∈𝑃 𝛿 ( ∑	|g 𝐾 | 2	) 𝑘	(A.2)

𝑘 ∏ 𝑗=1 ∫

20417942, 2024, 1, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12231 by Université de Saint-Etienne (Jean Monnet)/Library, Wiley Online Library on [12/11/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Funding information EPSRC, Grant/Award Number: EP/V521917/1; NSF, Grant/Award Number: DMS-1902763; Ben Green's Simons Investigator Grant, Grant/Award Number: 376201; FWF-ANR, Grant/Award Numbers: I 4945-N, ANR-20-CE91-0006; Australian Research Council, Grant/Award Number: FT20010039; American Institute of Mathematics

Iterating this gives 𝑆(𝛿) 2𝑘 ⩽ (2 2𝑘-1 𝑘 2𝑘 ) 𝑁 𝑆( 𝛿 𝜅 𝑁 ) 2𝑘 + 𝑁2 2𝑘-1 𝜅 -(4𝑘-2) (𝑞𝜅) -𝑘(𝑘-1) for all positive integers 𝑁 for which 𝜅 𝑁 ⩾ 𝛿; in particular, applying this with 𝑁 = ⌊ log 𝛿 -1 log 𝜅 -1 ⌋, and noting that 𝑆(𝛿∕𝜅 𝑁 ) ⩽ (𝛿∕𝜅 𝑁 ) -1∕2 ⩽ 𝜅 -1∕2 , we have 1) .

By choosing 𝜅 = 𝜅(𝜀) sufficiently small so that

⩽ 2𝑘𝜀, one obtains 𝑆(𝛿) ≲ 𝜀 𝛿 -𝜀 , as desired.


M J Malaska

R M C Lopes

A Hayes

Schoenfeld

T Verlander

M Florence

S Birch

A Le Gall

A Solomonidou

J Radebaugh

R Lorenz

AN UPDATED ORGANIC INVENTORY ESTIMATE FOR TITAN

Introduction: Saturn's moon Titan is an organic chemist's paradise. High in Titan's atmosphere, complex organic molecules are constructed from photochemical reactions involving methane and nitrogen. These products eventally make their way down to the surface where they create organic landforms that cover Titan's icy crust. With the completion of a global map of Titan's surface [START_REF] Lopes | Nature Astronomy[END_REF], as well as characterization of the different geological terrain units [START_REF] Malaska | [END_REF] we reexamined the amount of organics on Titan surface originally estimated by estimated by Lorenz et al., (2009) [3].

Given accurate photochemical production rates, the total inventory of organics on Titan can help constrain the amount of time that Titan's organic factory has been producing complex organic moleucles. The expected vs. observed ratio of solid to liquid materials can be also used to infer hidden reservoirs of organics on Titan. Methods: Using a set of Synthetic Aperture Radar (SAR) mosaics from the Cassini mission through the final T126 radar pass, a global geomorphological map of Titan was created using ArcGIS functionalities. The different terrain classes were characterized by radar backscatter, microwave emissivity, and infrared surface spectra collected by Cassini's Visual and Infrared Mapping Spectrometer (VIMS) instrument. In general, high emissivity terrains are more consistent with organic materials, since water ice generally has a longer microwave pathlength,increasing chances for volume scattering. From our mapping, the surface area of the terrain types could be multiplied by an estimated average organic thickness (effective thickness) across the surface in order to determine the volume of the organic deposit contained in each terrain unit. In particular, this work constrained of the amount of organics contained in the labyrinth terrains.

Results: The major terrains classes and types consistent with organic materials include: undifferentiated plains, dune sediments, labyrinth terrains, and lakes.

Undifferentiated Plains: Titan's vast undifferentiated plains have high emissivity and are thus likely composed of organic material [4], although VIMS suggests a surficial coating of -OH bearing materials in some of the high-latitude plains [5]. The undifferentiated plains cover 17% of Titan's surface, or 1.4E7 km 2 . While the exact depth of this deposit is unknown, Lopes et al. (2016) suggested that the smooth topography is suggestive of a thickness larger than 25 m thick [4]. Since microwaves can interrogate up to 1 m deep [6], we used 1 m as an absolute minimum thickness of the organic deposit. Due to their extensive surface area and potential thickness, the undifferentiated plains of Titan are a major driver of Titan's total organic inventory.

Dune sediments. Microwave emissivity and VIMS spectra are both consistent with the dune sediments being composed of organic materials [3,[7][8]. The amount of organic material was estimated by Rodriguez et al., (2014) [8] to be from 1.7 -4.4E5 km 3 .

Labyrinth terrains. Labyrinths are thick, dissected plateaux found at a variety of latitudes [9]. The plateaux, valley fill, and areas surrounding the labyrinth terrains, likely deposits from erosive removal of the plateaux, are all consistent with organic materials (Fig. 1). We measured the amount of dissection and the plateaux height from the regional base in order to determine the amount of plateau remaining, as well as the amount removed. For this exercise, we inventoried only the amount of organic plateau remaining. The average plateau height was 215 m. For those few labyrinths (ca 9% by labyrinth area) where we could not derive height information, we 1711.pdf 50th Lunar and Planetary Science Conference 2019 (LPI Contrib. No. 2132) picked a conservative height of 100 m. From this calculation, the total volume of material contained in the plateaux is 1.3E5 km 3 , this is more than all the organics contained in the visible filled lakes.

Lakes. Titan's filled hydrocarbon lakes have a total surface area of 9.04E5 km 2 and an estimated liquid volume of 0.7E5 km 3 [10]. Assuming that roughly 10% of the lake liquid volume is composed of dissolved nitrogen [11], this would leave a remnant amount of liquid hydrocarbons (and of that, most of it would be methane) of 0.63E5 km 3 .

Updated organic inventory estimate: For each terrain unit, the measured surface area, percent of the Titan total surface area, effective estimated thickness, and volume are shown in Table 1. We also include the percentage of the total organic volume based on a "rich" organic deposit scenario with 25 m thick undifferentiated plains scenario and 30 m thick average dunes and dune sediments, and a "lean" organic deposit scenario with 1 m thick undifferentiated plains and 12 m thick average dune sediment deposit. For the "rich" scenario, the sum total organic deposit of organics on Titan is 9.9E5 km 3 material, while for the lean scenario, the total is 3.8E5 km 3 total organic material. This corresponds to an average global layer of 12 m and 6 m respectively. If the liquids are removed from the estimate, the solids would account for 11 and 4 m thickness of organic material. A major reservoir of solid organic materials are the labyrinth terrains; although they only comprise <2% of the surface area, their thicknesss allows them to hold 13-35% of Titan's estimated organic inventory.

What we might have missed: The absolute thickness of the undifferentiated plans is unknown; the muted topography in the mid-latitude plains suggests the average layer could be significantly higher than 25 m thick. The amount of organics in the variable featured plains and in the scalloped plains is not estimated. Both of these units have large areal coverage, so even some incorporation of organics into these units would significantly add to the total amount of organic material. Discovery of additional labyrinth terrains in non-SAR imaged areas, likely at higher latitudes, could also increase the volume contribution from this unit. A porous regolith [12][13] could hold significant amounts of Titan's organic liquids in pore spaces; a pore-satured subsurface was invoked for thrust belt lubrication on Titan [14]. If these subsurface reservoirs have a higher ethane or propane content, they could also hold significant amounts of dissolved organic solids. Crustal clathrate reservoirs could also hold smaller organic molecules [15][16][17]. Finally, organics that were transported deeper into the crust, or even all the way to the subsurface ocean, perhaps at an earlier thin-crust epoch, would not be accounted for in our surface mapping.

Fig. 1 .

 1 Fig.1. SAR image of the southwest portion of Sikun Labyrinthus. Radar illumination is from the left, north is towards the top. Image is centered at (78°S, 32°W). The plateaux, valley fill, and empty lake basin (at center) are all consistent with a composition of organic materials with limited amounts of water ice.

50th Lunar and Planetary Science Conference 2019 (LPI Contrib. No. 2132)

Acknowledgements: This research was partly supported by the NASA Astrobiology Institute through its JPL-led team entitled "Habitability of Hydrocarbon Worlds: Titan and Beyond" and CDAP grant #NN13D466T. This work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. Government sponsorship is acknowledged.