Julien Trevisan

Limit laws in the lattice problem. II. The case of ovals

Nous étudions l'erreur du nombre de points d'un réseau unimodulaire qui tombent dans une ellipse dilaté et centré sur 0. On montre d'abord que l'étude de l'erreur, lorsqu'elle est normalisée par √ t où t est le paramètre de dilatation de l'ellipse, lorsque ce paramètre tend vers l'infini et lorsque le réseau considéré est aléatoire, se ramène à l'étude d'une transformée de Siegel S(f t )(L) qui dépend de t. Ensuite, on se ramène à l'étude du comportement asymptotic d'une transformée de Siegel avec poids aléatoires, S(F )(θ, L) où θ est un second paramètre aléatoire. Enfin, on montre que cette dernière quantité converge presque sûrement et on étudie l'existence des moments de sa loi.

Introduction

The lattice point problem is an open problem in the Geometry of Numbers, at least since Carl Friedrich Gauss took interest in which became the famous Gauss circle problem. The general problem states as followed. Let d be an integer greater than 1. We recall the following definition :

Definition 1. A subset L of R d is a lattice if it is a subgroup of R d such that L is discrete and span(L) = R d .
Let P be a measurable subset of R d of non-zero finite Lebesgue measure. We want to evaluate the following cardinal number when t → ∞ :

N (tP + X, L) = |(tP + X) ∩ L|
where X ∈ R d , L is a lattice of R d and tP +X denotes the set P dilated by a factor t relatively to 0 and then translated by the vector X. Under mild regularity conditions on the set P , one can show that :

N (tP + X, L) = t d Vol(P ) Covol(L) + o(t d )
where o(f (t)) denotes a quantity such that, when divided by f (t), it goes to 0 when t → ∞.

We are interested in the error term R(tP + X, L) = N (tP + X, L) -t d Vol(P ) Covol(L) .

In the case where d = 2 and where P is the unit disk D 2 , Hardy's conjecture in [START_REF] Hardy | The average order of the arithmetical functions p (x) and δ (x)[END_REF] stipulates that we should have for all > 0,

R(tD 2 , Z 2 ) = O(t 1 2 + ).
One of the result in this direction has been established by Iwaniec and Mozzochi in [START_REF] Iwaniec | On the divisor and circle problems[END_REF]. They have proven that for all > 0, R(tD 2 , Z 2 ) = O(t

7 11 + ).
This result has been recently improved by Huxley in [START_REF] Martin | Exponential sums and lattice points iii[END_REF]. Indeed, he has proven that :

R(tD 2 , Z 2 ) = O(t K log(t) Λ )
where K = 131 208 and Λ = 18627 8320 . In dimension 3, Heath-Brown has proven in [START_REF] Heath-Brown | Lattice points in the sphere[END_REF] that : R(tD 3 , Z 3 ) = O(t 21 16 + ).

These last two results are all based on estimating what are called exponential sums. Furthermore, in both cases, the error is considered in a deterministic way. Another approach was followed by Bleher, Cheng, Dyson and Lebowitz in [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF]. They took interest in the case where the dilatation parameter t is random. More precisely, they took interest in the case where t is being distributed according to the measure ρ( t T )dt (that is absolutely continuous relatively to Lebesgue measure) and where ρ is a probability density on [0, 1] and T is parameter that goes to infinity. In that case, they showed the following result :

Theorem ([4]). There exists a probability density p on R such that for every piecewise continuous and bounded function g : R -→ R,

lim T →∞ 1 T T 0 g( R(tD 2 , Z 2 ) √ t )ρ( t T )dt = R g(x)p(x)dx.
Furthermore p can be extended as an analytic function over C and verifies that for every > 0, p(x) = O(e -|x| 4-) when x ∈ R and when |x| → ∞.

Since the works of Kesten [START_REF] Kesten | Uniform distribution mod 1[END_REF] and [START_REF] Kesten | Uniform distribution mod 1 (ii)[END_REF], a lot of work has been done in estimating errors terms : we can cite, for example, [START_REF] Björklund | Counting in generic lattices and higher rank actions[END_REF] and [START_REF] Dolgopyat | Deviations of ergodic sums for toral translations i. convex bodies[END_REF]. Following their approaches, we have taken interest in what is happening to R when L is a random unimodular lattice, when P = E is an ellipse centred on 0 and when t is a deterministic parameter that goes to infinity. Let's recall the following definition :

Definition 2. The coarea of a lattice L of R 2 is the Lebesgue measure of a measurable fundamental set. Furthermore, a lattice is said to be unimodular if its corea is equal to 1.

We denote by S 2 the space of unimodular lattices and it can be seen as the quotient space SL 2 (R)/SL 2 (Z). We denote by µ 2 the unique Haar probability measure on it. Let us set :

Π = {(k 1 , k 2 ) ∈ Z 2 | k 1 ∧ k 2 = 1, k 1 0}
where we agree that if k 1 = 0, k 1 ∧ k 2 = 1 means that k 2 = 1. Let's set :

T ∞ = (T 1 ) Π where T 1 = R/Z and let's call λ ∞ the normalized Lebesgue measure product over T ∞ . Let's recall other usual definitions : Definition 3. For a lattice L ∈ S 2 , we say that a vector of L is prime if it not a non-trivial integer multiple of another vector of L. In fact, a vector l ∈ L is prime if, and only if, M l ∈ M L with M ∈ SL 2 (R) is prime and a vector (k 1 , k 2 ) ∈ Z 2 is prime if, and only if,

k 1 ∧ k 2 = 1.

Definition 4.

For every i ∈ {1, 2}, we call

L i = min{r > 0 | B f (0, r) contains i vectors of L lineary independent}
where B f (0, r) is the closed centred ball on 0 for the norm • of radius r. In fact, for almost all L ∈ S 2 , L 2 > L 1 and there exists only one couple of vectors (e 1 (L), e 2 (L)) such that (e 1 (L))

1 > 0, e 1 (L) = L 1 , (e 2 (L)) 1 > 0 and e 2 (L) = L 2 .
With these notations, one has that e ∈ L is prime if, and only if, e can be written as e = k 1 e 1 (L) + k 2 e 2 (L) with k 1 ∧ k 2 = 1. Definition 5. We call P + (L) the set of vectors e of L such that k 1 0 where e = k 1 e 1 (L) + k 2 e 2 (L) and where k 1 ∧ k 2 = 1. It is a set that is well-defined for almost all, relatively to µ 2 , L ∈ S 2 and all the vectors of P + (L) are prime vectors according to the previous remark. Definition 6. We say that a real random variable Z is symmetrical if

P Z = P -Z .
Let μ2 be a probability measure that has a smooth bounded density σ with respect to µ 2 . Let E be an ellipse centred around 0. We denote by • the usual euclidean norm over R 2 . The main result of this article is the following theorem : Theorem 1. For every real numbers a < b,

lim t→∞ μ2 R(tE, L) √ t ∈ [a, b] = (λ ∞ × μ2 ) (S(θ, L) ∈ [a, b]) where θ = (θ e ) ∈ T ∞ , S(θ, L) = 2 π e∈P + (L) φ(θ e ) e 3 2 with φ(θ e ) = m 1 cos(2πmθ e -3π 4 ) m 3 2
.

(1)

Furthermore, S(θ, L) :

• converges almost surely

• is symmetrical and its expectation is equal to 0

• admits a moment of order 1 + κ for any 0 κ < 1 3

• does not admit a moment of order 4 3 when σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0.

• when there exists α > 0 such that μ2 ({L ∈ S 2 | L 1 < α}) = 0 then S(θ, L) admits moments of all order 1 p < ∞.

The proof of Theorem 1 can be reduced to the case E = D 2 . Indeed, since the parameter of dilatation goes to infinity, we can replace it by t Area(E) and so the study is reduced to the case where E is of area 1. Furthermore, we note that in that case there exists M ∈ SL 2 (R) such that M E = D 2 and so the study is reduced to the case where P = D 2 by considering that L is distributed according to the probability measure (M -1 ) * μ2 . As a consequence, for the rest of the article, we are going to suppose

E = D 2 .
We notice that, in the case where σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0, the tail of the distribution is by far bigger than the tail of the limit distribution of [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF]. Indeed, in our case, the limit distribution admits of order less than 4 3 but does not admit a moment of order larger than 4 3 . In the case of [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF], the limit distribution admits moment of all order 1 p < ∞. In fact, they have better : the decreasing of the tail is at least estimated by x α e -|x| 4-where α → ∞ and is an arbitrary positive parameter.

The next section is a preliminary dedicated to a summation formula. It will enable us to give a heuristic explanation of the approach that we will follow. At the end of the section, we will give the plan of the paper.

Heuristic explanation and plan of the proof

We are first going to recall a summation formula (about this subject, see also section 2 of [START_REF] Bleher | On the distribution of the number of lattice points inside a family of convex ovals[END_REF]). This summation formula is based on the Poisson formula. Yet it does not apply in the case treated by Bleher, nor in our case. In fact, it applies after having regularized the problem (see the beginning of section 3 for more details). So it will give us a good insight of the problem and that is the reason why we are going to recall this summation formula. After doing so, we will apply it to our case. It will finally enable us to give heuristic explanations and a detailed plan of the proof of Theorem 1. Let γ be a simple, smooth, closed, convex and with a positive curvature curve in R 2 such that (0, 0) lies inside γ. For Ω γ the domain enclosed by γ, we call 1 tΩγ the characteristic function of tΩ γ . Then one has that :

N (tΩ γ , Z 2 ) = n∈Z 2 1 tΩγ (n).
By the Poisson summation formula (yet, as we said earlier, it does not apply here because 1 tΩγ is not smooth enough and so there is a convergence problem), one has that :

N (tΩ γ , Z 2 ) = n∈Z 2 1 tΩγ (2πn)
with 1 tΩγ the Fourier transform of 1 tΩγ defined by

1 tΩγ (ξ) = R 2
e ixξ 1 tΩγ (x)dx.

Note that 1 tΩγ (0) = Area(tΩ γ ). So we obtain that :

R(tΩ γ , Z 2 ) = n∈Z 2 -{0} 1 tΩγ (2πn). (2)
We need some notations before moving forward. For ξ ∈ R 2 -{0}, let's call x γ (ξ) the point on γ where the outer normal to γ coincides with ξ ξ where ξ refers to the usual euclidean norm of R 2 . Let's call also ρ γ (ξ) denote the curvature radius of γ at x γ (ξ) and

Y γ (ξ) =< ξ, x γ (ξ) >
where < •, • > denotes the usual scalar product over R 2 . With these notations, Bleher in [START_REF] Bleher | On the distribution of the number of lattice points inside a family of convex ovals[END_REF] obtains heuristically by giving an asymptotic expression of 1 tΩγ and by using (2) :

R(tΩ γ , Z 2 ) √ t = 1 π n∈Z 2 -{0} ρ γ (n) cos(2πtY γ (n) -3π 4 ) n 3 2 + O( 1 t ) (3)
where

Y = O(f (t)) is a quantity such that |Y | C|f (t)| with C 0.
Before saying what this summation formula gives in our case we just need to recall the definition of a dual lattice : Definition 7. For L ∈ S 2 , its dual lattice L ⊥ is defined in the following way

L ⊥ = {x ∈ R 2 | ∀l ∈ L, < l, x >∈ Z}.
Furthermore, if M is a matrix in SL 2 (R) that represents L then (M -1 ) T is a matrix that represents L ⊥ where (M -1 ) T is the transpose matrix of M -1 .

With the previously exposed summation formula, in our case we get, after some calculations (that we will expose later) :

R(tD 2 , L) √ t = 1 π l∈L ⊥ -{0} cos(2πt l -3π 4 ) l 3 2 + O( 1 t )
where L ⊥ is the dual lattice of L. Let's recall the following definition :

Definition 8. Let f be a function on R 2 .
Let's define formally :

S(f )(L) = e∈L e prime f (e). S(f ) is called the Siegel transform of f . If f has a compact support, S(f ) is well-defined. Let's set f t (x) = 1 π cos(2πt x -3π 4 ) x 3 2 1 R 2 -{0} (x).
As a consequence of the summation formula, heuristically the asymptotic behaviour of R(tD 2 ,L)

√ t
is the same as the asymptotic behaviour of the following Siegel transform

S(f t )(L ⊥ ) = 1 π l∈L ⊥ -{0} cos(2πt l -3π 4 ) l 3 2 . ( 4
)
We can, and we will, study it when t → ∞ by replacing L ⊥ by L (because L is supposed to be random). By the way, here we see that the randomness on t in the case treated in [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF] is somewhat replaced by the randomness on l where l ∈ L -{0} with L random and distributed according to μ2 . Yet, in [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF] and in [START_REF] Bleher | On the distribution of the number of lattice points inside a family of convex ovals[END_REF], this randomness on t enables the authors to use results about what is called almost periodic function (developed in [START_REF] Samoilovitch | Almost periodic functions[END_REF] and in [START_REF] Heath-Brown | The distribution and moments of the error term in the dirichlet divisor problem[END_REF]). More precisely, the authors use the following theorem : for every F : R + -→ R, if for every > 0, there exists a trigonometric polynomial P such that lim sup

T →∞ 1 T T 0 min(1, |F (t) -P (t)|)
then F (t) admits an asymptotic distribution.

Here, in our case, we can not work in this framework (because of the difference of nature of the randomness). So, we have to take another approach and the approach we are going to follow is directly taken off [START_REF] Dolgopyat | Deviations of ergodic sums for toral translations i. convex bodies[END_REF].

We now give the main steps of the proof of Theorem 1.

First step. By regularizing the problem, we are going to show that the quantity S A,prime (L, t), for A > 0 a fixed parameter that is taken large enough, when t → ∞, is close, in probability, to R(tD 2 ,L)

√ t
where S A,prime (L, t) is defined by :

S A,prime (L, t) = 1 π l∈L ⊥ prime 0< l A 1 l 3 2 m∈N-{0} cos(2πtm l -3π 4 ) m 3 2 . ( 5
)
It is different from the right-hand side of equation ( 4) because, first, the sum has been cut and it has to be cut because of a problem of convergence of the right-hand side of equation (4) ; second, we take into account a phenomenon of multiplicity (if l appears in the sum, 2l is also going to appear). Bleher in [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF] did also take into account such a phenomenon. He did it in order to get independence at infinity, as it was also done in [START_REF] Dolgopyat | Deviations of ergodic sums for toral translations ii[END_REF], and we do that for the same goal. By using a simple parity argument, by using the remark that follows Definition 3 and by replacing L by L ⊥ , one has that :

S A,prime (L ⊥ , t) = 2 π k 1 ∧k 2 =1 k 1 0 k 1 e 1 (L)+k 2 e 2 (L) A φ(t k 1 e 1 (L) + k 2 e 2 (L) ) k 1 e 1 (L) + k 2 e 2 (L) 3 2 (6)
where the function φ was defined by the equation [START_REF] Samoilovitch | Almost periodic functions[END_REF]. We have done that so from this stage onwards we consider vectors of P + (L) with a fixed indexation (that does not depend on L). Furthermore, this indexation will be very useful for the second step (for more details, see section 4).

Second step. We will show that the family of variables (t k 1 e 1 (L) + k 2 e 2 (L) ), whose values are in R/Z, are, when t → ∞, independent from one another and are identically distributed according to the normalized Haar measure on R/Z. The idea here is basically the same as in [START_REF] Pavel | Distribution of the error term for the number of lattice points inside a shifted circle[END_REF] and in [START_REF] Heath-Brown | The distribution and moments of the error term in the dirichlet divisor problem[END_REF] where the respective authors used the fact that the square roots of square free integers are Z-free. In our case, to prove the result, we will decompose the space of unimodular lattices into small geodesic segments, calculate the Taylor series of k 1 e 1 (L) + k 2 e 2 (L) at order 1 on such a segment and show that the coefficients of order 1 are Z-free. We will also prove that these variables are independent, when t → ∞, from the variables L due to the presence of the factor t.

Third step. Thanks to the first and second step, we will see that the asymptotic distribution of R(tD 2 ,L)

√ t
is the distribution of

S(θ, L) = k=(k 1 ,k 2 ) k 1 ∧k 2 =1 k 1 0 φ(θ k ) k 1 e 1 (L) + k 2 e 2 (L) 3 2 (7)
under the assumption that the quantity S ( θ, L) is well-defined and where θ

= (θ k ) ∈ T ∞ is distributed according to λ ∞ , L ∈ S 2 is distributed according to μ2 It should be noted that the (φ(θ k )
) is a family of independent and identically distributed real random variables. Furthermore, all of them are symmetrical and have a compact support. The third step of the proof will consist in showing a general result. Let the Z k be real non-zero independent identically distributed real random variables from a probability space Ω ω that are symmetrical and have a compact support. Then we are going to show that

SA (ω, L) = k=(k 1 ,k 2 ) k 1 ∧k 2 =1 k 1 0 k 1 e 1 (L)+k 2 e 2 (L) A Z k (ω) k 1 e 1 (L) + k 2 e 2 (L) 3 2 (8)
converges almost surely when A → ∞. S looks like a Siegel transform but it is not because of the numerator of the considered terms. We are going to call it a modified Siegel transform with random weights. An object of this type was already studied in [START_REF] Dolgopyat | Deviations of ergodic sums for toral translations i. convex bodies[END_REF]. As a consequence, S ( θ, L), as the limit when A → ∞ of a particular S, will be well-defined. We will also study the existence of moments of the almost sure limit. In particular, we are going to see that the optimal κ is 1 3 (see statement of Theorem 1). Heuristically, it is because the magnitude of S is given by the term

Z 1,0 L 3 2 1 with Z 1,0 = 0.
After doing all of that, we will finally get the validity of Theorem 1.

Plan of the paper. The next section will be dedicated to deal with the first step of the proof, namely it will show that R(tD 2 ,L)

√ t
is close in probability with S A,prime (L ⊥ , t) when A is a fixed parameter taken large enough and t goes to infinity (see Proposition 1). We have to "cut" the sum because of the problem of convergence of the Fourier series of X -→ R(tD 2 +X,L)

√ t
which is due to the lack of regularity of the indicator function 1 tD 2 . To prove this, we are going to proceed by regularization which means here that we are going to smooth the indicator function 1 tD 2 via a Gaussian kernel.

In section 4, we tackle the second step of the proof, that is the fact that we have independence when t → ∞ of the (t k 1 e 1 (L) + k 2 e 2 (L) ), that they are identically distributed according to the normalized Haar measure over R/Z independent from L and so that, when t → ∞,

R(tD 2 ,L) √ t
has the same distribution of S(θ, L). In section 5, which is the last section, we are going to tackle with the third step of the proof, namely study the convergence of S(ω, L, A) when A → ∞ and the existence of moments of its limit. In the rest of the article, all the calculus of expectation E, of variance Var and of probability P will be made according to the measure μ2 . Furthermore, the expression typical is going to signify μ2 -almost surely.

Reduction to the study of the Siegel transform

The main object of this section is to show the following proposition : Proposition 1. For every α > 0, for every A > 0 large enough, for every t large enough, one has that :

P(∆ A,prime (L, t) α) α
where

∆ A,prime (L, t) = | R(tD 2 , L) √ t -S A,prime (L, t)| . ( 9
)
This proposition basically says that we can reduce the asymptotical study of R(tD 2 ,L)

√ t
to the study of its Fourier transform, taking into account a phenomenon of multiplicity. In fact, due to the triangle inequality, we only have to prove the following two lemmas : Lemma 1. For every α > 0, for every A > 0 large enough, for every t large enough, one has that :

P(∆ A (L, t) α) α
where

∆ A (L, t) = | R(tD 2 , L) √ t -H A (L, t)| with ( 10
)
H A (L, t) = 1 π l∈L ⊥ 0< l A cos(2πt l -3π 4 ) l 3 2 . ( 11
)
Lemma 2. For every α > 0, for every A > 0 large enough, for every t large enough, one has that :

P(|S A,prime (L, t) -H A (L, t)| α) α.
Proof of Proposition 1. One has that :

∆ A,prime (L, t) ∆ A (L, t) + |S A,prime (L, t) -H A (L, t)|. ( 12
)
The Lemma 1 and the Lemma 2 imply then the wanted result.

Let's say a few words about Lemmas 1 and 2 before following with their respective proofs. The Lemma 1 says that the study of R(tD 2 ,L) √ t can be reduced to the study of its Fourier transform. The Lemma 2 says that the phenomenon of multiplicity (the fact that for a prime vector l, 2l, 3l etc. appear in the sum H A (L, t) when A → ∞) is not so important. We only have to gather all the multiples of a prime vector (which corresponds to the infinite sum over m, see equation ( 5)), so that we focus on prime vectors.

Proof of Lemma 1

First, we are going to prove the Lemma 1. To do so, we are following closely the approach of [START_REF] Bleher | On the distribution of the number of lattice points inside a family of convex ovals[END_REF], yet with some differences because in our case it is not the radius of dilatation that is random but the lattice (or, equivalently and in a certain sense, the oval). For x ∈ R 2 and t > 0, let's define

λ(x; t) = t 2 4π e -t 2 4π x 2
and, for

M ∈ SL 2 (R), λ M (x; t) = λ(M x; t). ( 13
)
We recall that :

R 2 λ M (x; t)dx = 1 ( 14
)
and that the Fourier transform of λ M (•; t) can be expressed as

λ M (ξ; t) = e -(M -1 ) T ξ 2 t 2 . ( 15
)
We introduce the following function :

χ S 1 ,M (x; t) = (1 tΩ M -1 S 1 * λ M (•; t))(x) = R 2 1 tΩ M -1 S 1 (y)λ M (x -y; t)dy ( 16
)
(it is a regularization of the function χ tΩ M -1 S 1 ). Let us also set :

N reg (tD 2 , M ) = n∈Z 2 χ S 1 ,M (n; t) and (17)
(the index "reg" stands for regularized)

F (M, t) = N reg (tD 2 , M ) -Area(tD 2 ) √ t . ( 18
)
Let L be a unimodular lattice such that e 1 (L) and e 2 (L) are well-defined and let

M = [e 1 (L), e 2 (L)] if det([e 1 (L), e 2 (L)]) > 0 (19) and M = [e 2 (L), e 1 (L)] if det([e 2 (L), e 1 (L)]) > 0. ( 20
)
Then M is a matrix that represents L. Now, let's call :

∆ 1 (L, t) = | R(tD 2 , L) √ t -F (M, t)| and (∆ 2 ) A (L, t) = |F (M, t) -H A (L, t)| (21)
so one has that :

∆ A (L, t) ∆ 1 (L, t) + (∆ 2 ) A (L, t). ( 22
)
The proof of Lemma 1 lies on the two following lemmas :

Lemma 3. ∆ 1 (L, t) converges almost surely to 0 when t → ∞.
Lemma 4. For all α > 0, for all A large enough, for all t large enough,

P((∆ 2 ) A (L, t) α) α.
Proof of Lemma 1. It is the direct consequence of equation ( 22) and of Lemmas 3 and 4.

Lemma 3 basically tells us that the study of R(tD 2 ,L)

√ t
can be reduced to the study of one of its regularized Fourier series, whereas Lemma 4 means that the asymptotical study of this regularized Fourier series can be brought back to the study of the non-regularized Fourier series. The next subsubsection is dedicated to the proof of Lemma 3 and the subsubsection after it is dedicated to the proof of Lemma 4.

Proof of Lemma 3

The proof of Lemma 3 is based on two sublemmas. The first one is the following :

Sublemma 1. For all x ∈ R 2 , for all t > 0, |χ S 1 ,M (x; t) -1 tΩ M -1 S 1 (x)| e -t 2 4 dist(M x,tS 1 ) 2 where for all z ∈ R 2 , dist(z, tS 1 ) = inf y∈tS 1 |z -y|.
Proof. One has that :

|χ S 1 ,M (x; t) -1 tΩ M -1 S 1 (x)| = | y / ∈tD 2 t 2 4π e -M x-y 2 4 dy| if M x ∈ tD 2 and |χ S 1 ,M (x; t) -1 tΩ M -1 S 1 (x)| = | y∈tD 2 t 2 4π e -M x-y 2 4 dy| if M x / ∈ tD 2
because of equation ( 14) and by making the change of variable y = M u. The proof of Lemma 3.2 from [START_REF] Bleher | On the distribution of the number of lattice points inside a family of convex ovals[END_REF] gives the wanted result.

The second sublemma gives an estimate of dist(M n, tS 1 ). It states in the following way :

Sublemma 2.
For almost all L ∈ S 2 , for all t large enough, for all n ∈ Z 2 -{0}, we have that :

dist(M n, tS 1 ) Ct n where C = ( e 1 (L) + e 2 (L) ) L 1 .
We recall the reader that L 1 was defined in Definition 4.

Proof. By definition of dist, we have that :

dist(M n, tS 1 ) M n - M n M n t = |1 - t M n | M n . (23)
Furthermore, one has that :

M n M n ( 24
)
where M is the matrix norm subordinated to • . With the equation ( 19) and equation (20), one can see that :

M e 1 (L) + e 2 (L) . ( 25
)
Also, we have that :

|1 - t M n | t L 1 ( 26
)
for t large enough. By using the equations ( 23), ( 24), ( 25) and ( 26), one gets that :

dist(M n, tS 1 ) ( e 1 (L) + e 2 (L) ) t L 1 n . ( 27
)
Now we can prove Lemma 3.

Proof of Lemma 3. By using the equation ( 21), we have that :

∆ 1 (L, t) 1 √ t n∈Z 2 |χ S 1 ,M (n; t) -1 tΩ M -1 S 1 (n)|.
So, Sublemma 1 and Sublemma 2 give that :

∆ 1 (L, t) 1 √ t n∈Z 2 -{0} e -C 2 t 4 4 n 2 + e -t 4 4 √ t = O( 1 √ t ).

Proof of Lemma 4

To prove Lemma 4, we first need to give another expression of F (M, t), obtained via the Poisson formula. It is the object of the following lemma :

Lemma 5. F (M, t) = 1 π n∈Z 2 -{0} 1 (M -1 ) T n 3 2 cos(2πt (M -1 ) T n - 3π 4 ) λ M (2πn; t) + O M (t -1 )
where the M in index of O M is to signal that it depends on M (or, equivalently, on the lattice L).

To prove it, we first need a calculatory sublemma :

Sublemma 3. Let D ∈ SL 2 (R) and γ = DS 1 .
Then one has for every ξ ∈ R 2 -{0} :

x γ (ξ) = D D T ξ D T ξ , ρ γ (ξ) = ξ 3 D T ξ 3 and Y γ (ξ) = D T ξ
where D T is the transpose of the matrix D.

Proof. Let's set y = D -1 x γ (ξ) ∈ S 1 . One knows that :

k = y ( 28
)
where k is the outer normal to S 1 at y. Let's call T the unit tangent vector of S 1 such that (k, T ) is an orthonormal and direct basis of R 2 . Let's call R the rotation matrix of angle π 2 , so that

R = 0 1 -1 0
and so one has :

Rk = T . ( 29
)
By definition of x γ (ξ) and because D is a linear map such that det(D) > 0, one knows that DT is a tangent vector of γ at the point x γ (ξ) such that ( ξ ξ , DT ) is an orthogonal and direct basis of R 2 . By using (29), we now know that : (ξ, DRk) is an orthogonal and direct basis of R 2 . By using the fundamental property of the adjoint of an operator, we get that :

(R -1 D T ξ, k) is an orthogonal basis of R 2 . ( 30
)
So one gets that : R -1 D T ξ = aT where a is a non-zero real. Finally, we have

D T ξ = -ak (31)
because of (29).

The equation (28) and the fact that k is unitary give us that :

x γ (ξ) = ±D D T ξ D T ξ .
Yet, because (ξ, DRk) is a direct basis of R 2 we know that :

(ξ, ±DRD T ξ) is a direct basis.
Yet, one can see that RDR = -(D -1 ) T and so one gets that : (Rξ, ∓ξ) is a direct basis.

So, one gets finally that :

x γ (ξ) = D D T ξ D T ξ . ( 32
)
As a consequence, we get immediately that :

Y γ (ξ) = D T ξ .
Then by using the parametrization t -→ (cos(t), sin(t)) of the circle and the expression of a curvature radius when using a parametrization, one gets that at a point

X of γ = DS 1 , ρ(X) = (D -1 ) T D -1 X 3 .
So, with equation (32), finally one gets that :

ρ γ (ξ) = ξ 3 D T ξ 3 .
We can now tackle the proof of Lemma 5.

Proof of Lemma 5. According to the equation ( 18), the Poisson summation formula and because of the fact that 1tΩ M -1 S 1 (0) = Area(tΩ M -1 S 1 ) one has that :

F (M, t) = 1 √ t n∈Z 2 -{0} 1 tΩ M -1 S 1 (2πn) λ M (2πn; t). ( 33
)
Yet, according to Lemma 2.1 from [START_REF] Bleher | On the distribution of the number of lattice points inside a family of convex ovals[END_REF], one has that :

1 tΩ M -1 S 1 (ξ) = √ t ξ -3 2 ± 2πρ M -1 S 1 (±ξ) exp(±i(tY M -1 S 1 (±ξ) - 3π 4 )) + O M (t -1 2 ξ - 5 2 )
(34) By using Sublemma 3, we get with equation (33) :

F (M, t) = 1 π n∈Z 2 -{0} 1 (M -1 ) T n 3 2 cos(2πt (M -1 ) T n - 3π 4 ) λ M (2πn; t) + O M (t -1 ). ( 35
)
Using the equation ( 15), the equation ( 21), the fact that if M represents a lattice L, (M -1 ) T represents the dual lattice L ⊥ , and the previous lemma, that is Lemma 5, one gets that :

(∆ 2 ) A (L, t) ∆ 2,1 (L, t) + (∆ 2,2 ) A (L, t) + (∆ 2,3 ) A (L, t) (36) where ∆ 2,1 (L, t) = O M (t -1 ) (37) (∆ 2,2 ) A (L, t) = | l∈L ⊥ 0< l A 1 l 3 2 cos(2πt l - 3π 4 )(1 -e -(2π) 2 l 2 t 2 )| (38) (∆ 2,3 ) A (L, t) = | l∈L ⊥ A< l 1 l 3 2 cos(2πt l - 3π 4 )e -(2π) 2 l 2 t 2 |. (39)
So, if we prove the following lemmas, we will get Lemma 4 and, in fine, get Lemma 1 : Lemma 6. ∆ 2,1 (L, t) converges almost surely to 0 when t → ∞.

Let's remark, by the way, that this last lemma is immediate according to equation (37).

Lemma 7.

For all A > 0, (∆ 2,2 ) A (L, t) converges to 0 when t → ∞.

Lemma 8. For all α > 0, for all A large enough, for all t large enough, P((∆ 2,3 ) A (L, t) α) α.

Proof of Lemma 4. Let α > 0. Let's take A large enough so that for all t large enough,

P((∆ 2,3 ) A (L, t) α) α.
It is possible according to Lemma 8. According to Lemma 7, according to Lemma 6 and because the almost-sure convergence imply the convergence in probability, even if it means taking t larger, one can suppose that :

P((∆ 2,1 )(L, t) α) α and P((∆ 2,2 ) A (L, t) α) α.
By using equation 36, one gets the wanted result.

Before following with the proof of Lemma 1, let's say a few words about Lemma 7 and Lemma 8. The first tells us that the non-regularized Fourier series is "close" enough to the regularized Fourier series whereas the second one tells us that the large terms of the regularized Fourier series do not matter, in a certain sense, for our study. It remains only to prove Lemma 7 and Lemma 8. We will do just that in the next two subsubsections.

Proof of Lemma 7

Proof of Lemma 7. Let l ∈ L ⊥ . Then one has :

|1 -e -(2π) 2 l 2 t 2 | (2π) 2 l 2 t 2 . ( 40
)
With this equation and with equation (38), one gets that :

(∆ 2,2 ) A (L, t) l∈L ⊥ 0< l A (2π) 2 t 2 l 1 2 . ( 41
)
It follows that there exists C(L) > 0 such that : [START_REF] Vinogradov | Limiting distribution of visits of sereval rotations to shrinking intervals[END_REF], [START_REF] Kelmer | The second moment of the siegel transform in the space of symplectic lattices[END_REF]). For f a piecewise smooth function with compact support on R 2 , one has :

(∆ 2,2 ) A (L, t) C(L) A 5 2 t 2 . ( 42
• S 2 S(f )dµ 2 = c 1 R 2 f dλ • When f is even, (b) S 2 S(f ) 2 dµ 2 C R 2 f 2 dλ + c 2 ( R 2 f dλ) 2
where C > 0.

With this lemma, we are going to prove two lemmas that will enable us to prove Lemma 8 by using Chebyshev's inequality : the first one is intended to estimate the expectation of (∆ 2,3 ) A (L, t) to see that it goes to 0 when t → ∞ (uniformly in A), the second one is intended to estimate its variance to see that it can be as uniformly small in t as one wants if A is chosen large enough. Until the end of this section, we are going to suppose A > 1.

Lemma 10.

E((∆ 2,3 ) A (L, t)) = O( 1 t ).
Proof. One has :

(∆ 2,3 ) A (L, t) = l∈L ⊥ prime A< l f (l) (43)
where

f (l) = 1 l 3 2 k∈N-{0} cos(2kπt l -3π 4 ) k 3 2 e -(2kπ) 2 l 2 t 2 . ( 44
)
The Lemma 9 gives us then that :

E((∆ 2,3 ) A (L, t)) C R 2 f (x)1 x >A dx ( 45
)
where C > 0 because the density of μ2 is supposed to be bounded. By passing into polar coordinates, one gets that :

E((∆ 2,3 ) A (L, t)) C ∞ r=A k∈N-{0} cos(2kπtr-3π 4 ) k 3 2 e -(2kπ) 2 r 2 t 2 r 1 2 dr ( 46
)
(the constant C has changed but it does not matter). Lebesgue's dominated convergence theorem gives us that :

E((∆ 2,3 ) A (L, t)) C k∈N-{0} 1 k 3 2 ∞ r=A cos(2kπtr -3π 4 ) r 1 2 e -(2kπ) 2 r 2 t 2 dr. ( 47
)
Finally, an integration by part gives us that :

E((∆ 2,3 ) A (L, t)) C k∈N-{0} 1 k 3 2 ( - sin(2kπtA -3π 4 )e -(2kπ) 2 A 2 t 2 2kπtA 1 2 + ∞ A sin(2kπtA -3π 4 )e -(2kπ) 2 A 2 t 2 4kπtr 3 2 dr + ∞ A sin(2kπtr -3π 4 ) 2kπt (2πk) 2 t 2 2r 1 2 e -(2kπ) 2 r 2 t 2 dr). ( 48
)
By using that r

1 2
r (because A > 1), one has finally, by estimating the three terms of the right member :

E((∆ 2,3 ) A (L, t)) = O( 1 t ). Lemma 11. Var((∆ 2,3 ) A (L, t)) = O( 1 A )
where the O can be chosen independent from t.

Proof. By using the same notation as before, by using again the Lemma 9 and by using the Lemma 10, one gets that :

Var((∆ 2,3 ) A (L, t)) C R 2 f 2 (x)1 x >A dx. ( 49
)
So, by passing into polar coordinates, one gets that :

Var((∆ 2,3 ) A (L, t)) C2π ∞ r=A 1 r 2 ( k∈N-{0} cos(2kπtr -3π 4 ) k 3 2 e -(2kπ) 2 r 2 t 2 ) 2 dr. ( 50
) Because ∞ k=1 1 k 3 2
< ∞, we get the wanted result.

We can now prove the Lemma 8.

Proof of Lemma 8. The Chebyshev's inequality gives the wanted result if, first, we choose A large enough and, second, we choose t large enough so that E((∆ 2,3 ) A (L, t)) and Var((∆ 2,3 ) A (L, t)) are small enough. These choices are possible according to Lemmas 10 and 11.

So, now the proof of Lemma 1 is complete and we will conclude this section by proving the Lemma 2 so that the proof of Proposition 1 will be complete.

Proof of Lemma 2

To prove the Lemma 2, we are going to take the same kind of approach as before : estimate the expectation and the variance of the quantity S A -S A,prime and get the result via the Chebyshev's inequality. We have that :

H A (L, t) -S A,prime (L, t) = l∈L ⊥ prime f (l) (51)
where

f (l) = 1 l 3 2 1 0< l A k A l +1 cos(2kπt l -3π 4 ) k 3 2 . ( 52
)
With this expression, we see that we are going to have a little problem of integrability at 0 when using the Lemma 9. That's why, we have to exclude 0 and we will suppose that L is chosen so that L ⊥ 1 where 0 < < 1. Only a small number of lattices is excluded according to this lemma that we recall : Lemma 12. For every 0 < < 1, one has that

P( L 1 < ) = O( 2 ).
Proof. It is a consequence of Lemma 9 by taking

S(f )(L) = l∈L 1 B f (0, ) (l)
where 1 B f (0, ) (l) is the indicator function of the closed ball for the norm centred on 0 of radius .

Thus, for the chosen lattices, we have :

H A (L, t) -S A,prime (L, t) = l∈L ⊥ prime f (l)1 l = ∆ 3, ,A,t (L). ( 53
)
Lemma 13.

E(∆ 3, ,A,t (L)) = O ,A ( 1 t ).
Proof. By using the Lemma 9, one gets that :

E(∆ 3, ,A,t (L)) C A 1 r 1 2 k A r cos(2kπtr -3π 4 ) k 3 2
dr.

(54)

Lebesgue's dominated convergence theorem gives us that :

A 1 r 1 2 k A r cos(2kπtr -3π 4 ) k 3 2 dr = ∞ k=1 1 k 3 2 A max( A k , ) cos(2kπtr -3π 4 ) r 1 2 dr. ( 55
)
An integration by part as in the proof of lemma 10 and the equation (54) give us finally that :

E(∆ 3, ,A,t (L)) = O ,A ( 1 t ). ( 56
)
Lemma 14. There exists K > 0 such that :

Var(∆ 3, ,A,t (L)) K(- log( ) A + log(A) A ).
Proof. Lemma 9 gives us that :

Var(∆ 3, ,A,t (L)) C2π A 1 r 2 ( k A r cos(2πktr -3π 4 ) k 3 2
)2 dr.

(57)

Yet one also has that for all x > 0 :

k x 1 k 3 2 D x 1 2 (58)
where D > 0. Thus, equations ( 57) and (58) imply that :

Var(∆ 3, ,A,t (L)) 2πCD A A 1 r dr. ( 59
)
We can now give the proof of Lemma 2.

Proof of Lemma 2. First we take 1 > > 0 small enough so that the measure of the neglected lattices, id est the lattices such that L 1 < , is small enough. It is possible according to Lemma 12. Then we take A large enough so that Var(∆ 3, ,A,t (L)) is small enough. It is possible according to Lemma 14. Finally, we take t large enough so that E(∆ 3, ,A,t (L)) is small enough, which is possible according to Lemma 13, and conclude by using Chebyshev's inequality.

So, we are now brought back to the study of S A,prime (L, t) when t → ∞ and the next section is dedicated to it. We are going to replace L ⊥ by L (it changes nothing because we are studying the asymptotic convergence in law with L ∈ S 2 distributed according to μ2 ).

4 Study of S A,prime (L, t) when t → ∞

Reductions for the study of S A,prime (L, t) and proof of Theorem 1

Before entering in the main object of this section, we need to do a small rewriting of S A,prime (L, t). We recall that a vector l ∈ L is prime if, and only if, Kl ∈ KL is prime where K ∈ SL 2 (R). Furthermore, a vector (l 1 , l 2 ) ∈ Z 2 is prime if, and only if, l 1 ∧ l 2 = 1. By using the symmetry l -→ -l, we deduce that S A,prime (L, t) can be rewritten as followed :

S A,prime (L, t) = 2 π k∈Π A (L) Z k (L, t) Y k (L) (60)
where, for k

= (k 1 , k 2 ) ∈ Z 2 , Y k (L) = k 1 e 1 (L) + k 2 e 2 (L) Z k (L, t) = m∈N-{0} cos(2πtm k 1 e 1 (L) + k 2 e 2 (L) -3π 4 ) m 3 2 , ( 62
)
and where

Π A (L) = {(k 1 , k 2 ) ∈ Z 2 | k 1 ∧ k 2 = 1, k 1 0 k 1 e 1 (L) + k 2 e 2 (L) A} (63)
and here we agree that if (k 1 , k 2 ) ∈ Π A (L) then k 1 = 0 implies that k 2 = 1 (for the definition of e 1 (L) and e 2 (L) see Definition 4).

Let's recall that :

Π = {(k 1 , k 2 ) ∈ Z 2 | k 1 ∧ k 2 = 1, k 1 0}. ( 64
)
Our goal now is to prove the following proposition :

Proposition 2. {Z k (L, t)} k∈Π converge, when t → ∞, towards independent identically distributed real random variables that have a compact support, are symmetrical and are independent of L.

In the next section we are going to consider the sums of the type

SA (ω, L) = k∈Π A (L) Z k (ω) k 1 e 1 (L) + k 2 e 2 (L) 3 2
where Z k are non-zero real independent identically distributed random variables from Ω ω that are symmetrical and have a compact support. We will see, with Proposition 7, that the sums of this type :

• converge almost surely (on the space Ω × S 2 )

• their respective limits are symmetrical and their expectations are equal to 0

• their respective limits admit moment of order 1 + κ for any 0 κ < 1 3

• their respective limits do not admit a moment of order 4 3 when σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0.

• when there exists α > 0 such that μ2 ({L ∈ S 2 | L 1 < α}) = 0 then their respective limits admit moments of all order 1 p < ∞.

We are going to see now that it is enough to prove Proposition 2 and Proposition 7 to establish Theorem 1, with the exception of the exact form of the limiting law (yet it is given by Proposition 3).

Proof of Theorem 1. Let ψ ∈ C ∞ c (R). Let > 0.
According to Proposition 1, we can take A as large as we want and then t as large as we want so that :

|E ψ( R(tD 2 , L) √ t ) -E (ψ(S A,prime (L, t))) | . ( 65
)
Furthermore, even if it means neglecting lattices that form a set of small measure, one can suppose that only a finite number of k, independent of L, intervene in the equation (60). Thus, thanks to 2, one has that :

|E (ψ(S A,prime (L, t))) -E ψ( SA (ω, L)) | (66)
where the Z k (ω) in SA (ω, L) are given by Proposition 2. Furthermore, Proposition 7 gives us that :

|E ψ( SA (ω, L)) -E ψ( lim A→∞ SA (ω, L)) | (67)
with lim A→∞ SA (ω, L)) that verify all the listed properties. So, equations (65), ( 66) and (67) give the wanted result.

The main reason why Z k are going to be independent from L is the presence of the factor t.

The main reasons why the rest of Proposition 2 will be true are the presence of the factor t in Z k and the fact that the coefficients of order 1 of the Taylor series of ( k 1 e 1 (L)+k 2 e 2 (L) ) k∈Π A (L) on a small geodesic segment are Z-free.

In order to prove Proposition 2, it is actually enough to prove the following proposition :

Proposition 3. For k = (k 1 , k 2 ), let θ k (L, t) = t k 1 e 1 (L) + k 2 e 2 (L) mod 1. ( 68
)
Then, we have that {θ k (L, t)} k∈Π converge, when t → ∞, towards random variable that are independent identically distributed, are distributed according to the Lebesgue measure λ over R/Z and are independent from L.

Thanks to this proposition, we now understand why the limit law of R(tD 2 ,L) √ t is given by S(θ, L). To prove this last proposition, it is sufficient to prove the following proposition where e(θ) stands for exp(i2πθ) :

Proposition 4. For every l ∈ N -{0}, for every ψ ∈ C ∞ c (S 2 ), for every (p 1 , • • • , p l ) ∈ Z l -{0}, one has : E ψ(L)e( l h=1 p h θ k h ) → t→∞ 0 ( 69
)
where the k h ∈ Π are all distinct.

Before passing to the proof of Proposition 4, let's give some heuristic about it. Basically, by working with a foliation of the space S 2 given by small enough geodesic segments, we are first going to have :

E ψ(L)e( l h=1 p h θ k h ) ≈ E(ψ)E e( l h=1 p h θ k h )
due to the presence of the factor t in θ.

The right member will go to 0 when t goes to infinity because a Riemann-Lebesgue lemma will apply because quantities "close" to the variables θ k are typically Z-free (see the heuristic explanation of the second step). The rest of this section is now dedicated to the proof of the Proposition 4.

Foliation and local estimates

We recall that a foliation of the space S 2 is given by the orbits of the group δ where

δ(λ) = λ 0 0 1 λ .
To prove Proposition 4, we are going to look at what it is happening on a small "segment" of the form

S (L) = {δ(λ)L | λ ∈ [ 1 1 + , 1 + ]} (70)
where L ∈ S 2 and > 0 can be taken as small as possible. More precisely, we are going to show, when t → ∞, the independence of the (θ k ) and of L over smalls segments of the form S (L), as well as the fact that the (θ k ) are identically distributed and distributed according to the normalized Haar measure over R/Z. Let's set, for k = (k 1 , k 2 ) ∈ Π :

W k (L) = 2 j=1 k 2 j (e j (L)) 2 1 -(e j (L)) 2 2 k 1 e 1 (L) + k 2 e 2 (L) . ( 71
)
On a segment of the form S (L), the following lemma basically tells us how we can estimate the quantities Z k :

Lemma 15. For a typical L ∈ S 2 , there exists > 0 small enough such that for every λ ∈ [ 1 1+ , 1 + ], e 1 (δ(λ)L) = δ(λ)e 1 (L) and e 2 (δ(λ)L) = δ(λ)e 2 (L).

Furthermore, for such a λ, for k = (k 1 , k 2 ) ∈ Π, we have for h = λ -1,

k 1 e 1 (δ(λ)L) + k 2 e 2 ((δ(λ)L) = k 1 e 1 (L) + k 2 e 2 (L) (72)
+ W k (L)h + O k 1 ,k 2 ,L (h 2 ).
Proof. First, if the first part of the lemma is acquired, then the following equation are acquired via a basic calculation of Taylor series. Thus, we just have to prove the first part of the Lemma 15. Now let's prove that for a > 0 small enough, for every λ ∈ [ 1 1+ , 1 + ], e 1 (δ(λ)L) = δ λ e 1 (L). We note that for a typical lattice L, we have that

min l∈L-{±e 1 (L),±e 2 ,0} l e 2 (L) > e 1 (L) . ( 73
)
So there exists κ > 0 such that for every k ∈ L -{±e 1 (L), ±e 2 (L), 0},

k e 1 (L) + κ. ( 74
)
This last equation and the continuity of give that, for a > 0 small enough, for every

λ ∈ [ 1 1+ , 1 + ] : δ(λ)k δ(λ)e 1 (L) + κ 2 . ( 75
)
This last equation and the fact (δ(λ)e 1 (L)) 1 > 0 give us that δ(λ)e 1 (L) = e 1 (δ(λ)L). Finally, let's prove that, even if it means reducing > 0, for every λ ∈ [ 1 1+ , 1 + ], e 2 (δ(λ)L) = δ λ e 2 (L). We have the wanted result when the first inequality in the equation ( 73) is strict (by reasoning the same way as before). So let's suppose that this last inequality is an equality. Let's call l such that l 1 0 and such that e 2 (L) = l . If l 1 > 0 it means that the couple (e 1 (L), e 2 (L)) is not well-defined. If l 1 = 0 then L belongs to a negligible set according to the Lemma 4.5 from the article [START_REF] Skriganov | Ergodic theory on sl(n), diophantine approximations and anomalies in the lattice point problem[END_REF].

To prove Proposition 4 we see, in light of Lemma 15, that it would be convenient to prove the following proposition : Proposition 5. For a typical L ∈ S 2 , for every m ∈ N -{0}, for every family

(p 1 , • • • , p m ) ∈ Z m , for every k 1 , • • • , k m ∈ Π, all distinct if m i=1 p i W k i (L) = 0 (76) then p 1 = • • • = p m = 0.
In other words, for a typical L ∈ S 2 ,

(W k i (L)) i 1 is a Z-free family.
The next subsection is dedicated to prove this proposition.

Proof of Proposition 5

Let us set : With theses notations, one has :

X 1 = e 1 (L) 2 , ( 77
)
X 2 =< e 1 (L), e 2 (L) > , ( 78
)
X 3 = e 2 (L)
Lemma 16. • X 1 X 3 = 1 + X 2 2 • X 2 = ± cos(β) sin(β) Proposition 6. For a typical L ∈ S 2 , for all A k 1 , • • • , A kn , A h 1 , • • • , A hr where k 1 , • • • , k n and h 1 , • • • , h r
are all distinct from one another and from (1, 0), the square root

r i=1 A h i does not belong to the field Q(X 1 , X 2 , X 4 )[ A k 1 , • • • , A kn ]
with n 0, r 1. Consequently, the family formed by the square roots of product of different A k are Q(X 1 , X 2 , X 4 )-free.

We must exclude the case (1, 0) because A (1,0) = X 2 1 . Yet, in the proof of the Proposition 5, there won't be any major difficulties created because of this last fact. This proposition is an analogue of a lemma that can be used to prove that the family ( √ p) p∈P , where P is the set of prime numbers, is Z-free. The Proposition 6 will be proved by induction but to achieve it we first need a lemma. Lemma 17. For a typical lattice L ∈ S 2 , for all

P ∈ Q[Y 1 , Y 2 , Y 3 ], if P ( e 1 (L) 2 , cos(β) sin(β) , cos(α) sin(α) ) = 0 then P = 0.
As a consequence, for a typical lattice L ∈ S 2 , for all

P ∈ Q[Y 1 , Y 2 , Y 3 ], if P (X 1 , X 2 , X 4 ) = 0 then P = 0
where X 1 was defined by equation (77), X 2 was defined by equation (78) and X 4 was defined by equation (80).

Proof. The result is immediate if one has in mind that :

• A typical lattice L ∈ S 2 is determined by three, and only three, continuous parameters : α ∈]-π 2 , π 2 [-{0} which is the oriented angle from (1, 0) to e 1 (L), β ∈]-π 2 -α, π 2 -α[-{0} which is the oriented angle from e 1 (L) to e 2 (L) and r > 0 such that r 4 < 1 sin 2 (β) (see equation ( 81)) which is e 1 (L) . Yet we want to emphasize that all the choices of (α, β, r) ∈ {(α, β, r)

| α ∈]-π 2 , π 2 [-{0}, β ∈]-π 2 -α, π 2 -α[-{0}
, 0 < r and r 4 < 1 sin 2 (β) } does not give a lattice L ∈ S 2 : the set of (α, β, r) that suit are a submanifold of dimension 3.

• For a typical lattice L ∈ S 2 (like before), if there exists a P = 0 such that

P ( e 1 (L) 2 , cos(β) sin(β) , cos(α) sin(α) ) = 0
then L belongs to a countable union of submanifolds of dimension at most 2.

We can now prove the Proposition 6.

Proof of Proposition 6. We are first going to prove the first part of the proposition. Let's suppose that L is a typical lattice like in the Lemma 17. Let's prove that such a L is appropriate by making an induction on the parameter n ∈ N (r 1 is not fixed).

(Basis)

Let's suppose that a square root

r i=1 A h i belongs to Q(X 1 , X 2 , X 4 ). Then there exists P, Q ∈ Q[Y 1 , Y 2 , Y 4 ] such that P ∧ Q = 1 and such that : r i=1 A h i = ( P Q )(X 1 , X 2 , X 4 ). (89)
By squaring this equation and using Lemma 17, one gets the following equality between polynomials :

P 2 = Q 2 ( r i=1 A h i )(Y 1 , Y 2 ). (90)
So, we get that Q = 1. Furthermore, we have that for any h ∈ Π -{(1, 0)} :

A h (Y 1 , Y 2 ) = (h 1 Y 1 + h 2 Y 2 ) 2 + h 2 2 > 0 (
r i=1 A h i ∈ Q(X 1 , X 2 , X 4 )[ A k 1 , • • • , A kn , A k n+1 ]. Yet, we have : Q(X 1 , X 2 , X 4 )[ A k 1 , • • • , A kn , A k n+1 ] = U [ A k n+1 ] where U = Q(X 1 , X 2 , X 4 )[ A k 1 , • • • , A kn ].
So here there exist x, y ∈ U such that :

r i=1 A h i = x + y A k n+1 . ( 92
)
By squaring, one gets that :

r i=1 A h i = x 2 + y 2 A k n+1 + 2xy A k n+1 . (93)
Now, if x and y are different from 0, one has A k n+1 ∈ U which is excluded by the hypothesis of the induction. So y = 0 and in that case r i=1 A h i ∈ U , which is also excluded by the hypothesis of the induction, or x = 0 and then r i=1 A h i A k n+1 ∈ U , which is also excluded by the hypothesis of the induction. So, we obtain the first part of the proposition. Concerning the second part, let's enumerate the k that belong to Π and that are different from (1, 0). It forms a sequence (k n ) n 1 . Thanks to the first part of the proposition, we see that the sequence of fields (Q

(X 1 , X 2 , X 4 )( A k 1 , • • • , A kn )) is increasing for the inclusion. So, for all n 1, its dimension over Q(X 1 , X 2 , X 4 ) must be 2 n . Yet Q(X 1 , X 2 , X 4 )( A k 1 , • • • , A kn ) is generated by the product of square roots of A k 1 , • • • , A kn
where k i taken are all different from one another. But their numbers is equal to 2 n . So, they form a Q(X 1 , X 2 , X 4 )-free family.

We can now prove the Proposition 5.

Claim. With these notations, we have, for all L ∈ K 1 , that :

E ψe( l i=1 p i θ k i )|F 2 (L) = O( 2 ) + O( 1 at 2 ) + 1 a O( 2 ). (98)
To this end, let's set, for all > 0, δ( ) = 1 + -1 1 + .

Then one has for every L ∈ K 1 according to Lemma 15 :

E(ψe( l j=1 p j θ k j )|F 2 )(L) = 1 δ( 2 ) 2 1 1+ 2 -1   ψe( l j=1 p j θ k j )   (δ(h)L)dh = ψ(L) 1 δ( 2 ) 2 1 1+ 2 -1   e( l j=1 p j θ k j )   (δ(h)L)dh + O( 2 ) =   ψe( l j=1 p j θ k j )   (L) 1 δ( 2 ) 2 1 1+ 2 -1 e itD 1 (L)h+itD 2 (L,h) dh + O( 2 ) (99)
where

D 1 (L) = l j=1 p j W k j (L), (100)
D 2 (L, h) = l j=1 p j θ k j (δ(λ)L) - l j=1 p j θ k j (L) -D 1 (L)h such that (101) D 2 (L, h) = O(h 2 ) and (102) D 2 (L, •) is smooth around 0.
Thus, by integrating by part and by using equation (96), one gets that for all L ∈ K 1 :

1 δ( 2 ) 2 1 1+ 2 -1
e itD 1 (L)h+itD 2 (L,h) dh

= 1 δ( 2 )   
e itD 1 (L)h+itD 2 (L,h) itD 1 (L) Thanks to equation (98), the fact that µ 2 (K 1 ) 1 -1 and because of equation (97), we have that :

2 1 1+ 2 -1 + 1 D 1 (L)
|E   ψe( l j=1 p j θ k j )   | |E   ψe( l j=1 p j θ k j )1 K c 1   | + |E   ψe( l j=1 p j θ k j )1 K 1   | ψ ∞ 1 + O( 2 ) + O( 1 at 2 ) + 1 a O( 2 ). ( 104
)
By first choosing 1 > 0 small enough (note that a depends on 1 ), then choosing 2 > 0 and finally choosing t large enough, we obtain the wanted result.

We are now brought back to the study of the convergence, when A → ∞, of a sum of the type SA (ω, L) = where Z k are real non-zero independent and identically distributed random variables from Ω ω that are symmetrical and have a compact support. We are going to see in the next section that the sums of this type converge almost surely (on the space Ω × S 2 ) when A → ∞. Furthermore, we are going to see that the almost sure limit lim A→∞ SA (ω, L) :

• is symmetrical and its expectation is equal to 0

• admits moment of order 1 + κ for any 0 κ < 1 3

• does not admit a moment of order 4 3 when σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0.

• when there exists α > 0 such that μ2 ({L ∈ S 2 | L 1 < α}) = 0 then it admits moments of all order 1 p < ∞.

5 Asymptotic study of SA (ω, L) and proof of 1

The goal of this section is first to prove the following proposition :

Proposition 7. SA (ω, L) converges almost surely when A → ∞. Furthermore, we are going to see that the almost sure limit lim A→∞ SA (ω, L) :

• is symmetrical and its expectation is equal to 0

• admits moment of order 1 + κ for any 0 κ < 1 3

• does not admit a moment of order 4 3 when σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0.

• when there exists α > 0 such that μ2 ({L ∈ S 2 | L 1 < α}) = 0 then it admits moments of all order 1 p < ∞.

To prove this proposition, we first need some lemmas.

The following lemma basically gives us that the magnitude of lim A→∞ SA (ω, L) is given by Then there exists (M s ) s∈N-{0,1} , a sequence of positive real numbers (that does not depend on A), such that for all s 2:

E( k∈Π |H k,1 | s (ω, L, A, )) M s . ( 109
)
Furthermore, one has :

S ,2 (ω, L) = Z 1,0 L 3 2 1 1 L 1 < (110)
when > 0 is chosen small enough.

To prove this lemma, we need to recall a Theorem due to Minkowski.

Theorem 2 ([16]

). There exists K > 0 such that for all L ∈ S 2 ,

L 1 L 2 K 2 .
We can now prove Lemma 19.

Proof of Lemma 19. The first fact is a direct consequence of Lemma 9 and of the fact that the Z k are of compact support. Indeed, first let's say that it is enough to deal with the case μ2 = µ 2 . With the notations of Lemma 9, let's set

f (x) = 1 x 3 1 x A (111)
and S(f

)(L) = l prime ∈L f (l). ( 112
)
From the facts that the X k are independent, identically distributed, symmetrical with a compact support, one gets that :

E( k∈Π (B k,1,A ) 2 (ω, L, )) = E(Z 2 1,0 )E(F ). ( 113
)
The Lemma 9 gives then that :

E( k∈Π (B k,1,A ) 2 (ω, L, )) = E(Z 2 1,0 ) R 2 f (x)dx. ( 114
)
By passing into polar coordinates, one gets finally that :

E( k∈Π (B k,1,A ) 2 (ω, L, )) = 2πE(Z 2 1,0 )( 1 - 1 A ). ( 115
)
We do the same for the other k 3.

Concerning the second fact, if there exists (k 1 , k 2 ) ∈ Π such that k 1 e 1 (L) + k 2 e 2 (L) <

  β ∈] -π, π[-{0} is the oriented angle from e 1 (L) to e 2 (L) and where α is the oriented angle from (1, 0) to e 1 (L) (see the figure just after). So one has typically α ∈] -π 2 , π 2 [-{0} and β ∈] -π 2 -α, π 2 -α[-{0}.

1 (

 1 D 2 (L, •)) (h)e itD 1 (L)h+itD 2 (L,h) dh (99) and equation (103) give the wanted claim.

k=(k 1 ,k 2 )

 12 ∈Π A (L) Z k (ω) k 1 e 1 (L) + k 2 e 2 (L) 3 2

X 1, 0 L 1 . 2 1 2 1

 0122 Lemma 19. Let's set, for 0 < < 1 < A, B k,1,A (ω, L, ) = Z k (ω) k 1 e 1 (L) + k 2 e 2 (L) 3 A k 1 e 1 (L)+k 2 e 2 (L) , (105)B k,2 (ω, L, ) = Z k (ω) k 1 e 1 (L) + k 2 e 2 (L) 3 k 1 e 1 (L)+k 2 e 2 (L) < , (106) S ,1,A (ω, L) = k∈Π H k,1,A (ω, L, ) and(107)S ,2 (ω, L) = k∈Π H k,2 (ω, L, )(108)so that SA (ω, L) = S ,1,A (ω, L) + S ,2 (ω, L).

) 3 .1.4 Proof of Lemma 8

 3 To prove Lemma 8 we need to use what are called Siegel and Rogers formulas. Theses formulas will also be useful later in this paper. By setting c k = ζ(2) -k for k an integer larger than 1 and where ζ denotes the ζ function of Riemann, one has the following formulas :

	Lemma 9 ([15],

  91) and this inequality is true for everyY 1 , Y 2 ∈ R. So, every such A h is prime in the ring Q[Y 1 , Y 2 , Y 4 ]. Furthermore, the A h are not associated from one another, which means that for two different h 1 , h 2 , A h 1 does not divide A h 2 . The equation (90) leads us to an absurdity.

(Induction step) Let's suppose that the result is acquired for all k ∈ [0, n]. Furthermore, let's suppose that

,(61)

• For every k ∈ Π,

Proof. Because L ∈ S 2 , we have that

Furthermore, one has :

So by summing the equation (81) with the equation (82), one gets that :

The second item is also a direct consequence of equations ( 81) and (82).

The third item is obtained by using the fact that

The fourth item is obtained by using the two following equations :

Concerning the fifth and last item, thanks to the two following equations

we get that :

We conclude the proof of the fifth item by using the first item that has been proved at the beginning.

Now, let's set, for every k ∈ Π

Thanks to the second equation of Lemma 16, we have that :

and so, by multiplying, the equation ( 76) is equivalent to (with the same notations) :

At the end of the day, according to this last remark, to prove the Proposition 5, it is enough to prove the following proposition :

Proof of Proposition 5. Let's give us a typical L ∈ S 2 like in the Proposition 6 and like in the Lemma 17. Let's suppose also that :

with all the p i = 0 and with m 1. Then the equation ( 88) is verified. Yet, this equation means in particular that the family of the square roots of different

which is excluded by Proposition 6, except in the case where m = 1 or m = 2 and in this last case all k h must be equal to (1, 0). So, these two cases are also excluded in the end.

Proof of Proposition 4

Before starting the proof of Proposition 4, we only need a simple lemma :

all distinct, for every 0 < < 1, there exists a > 0 and K a measurable set of S 2 such that μ2 (S 2 -K ) and such that for all L ∈ K ,

Proof. It is a direct consequence of Proposition 5. Now, by using the foliation given by δ(λ) and previous results, we can now prove Proposition 4.

Proof of Proposition 4. The proof in all its generality can be made as in the case where μ2 = µ 2 . So, we will suppose for simplicity that μ2 = µ 2 .

Let

. For all > 0, we call F the tribe on S 2 generated by the S (L). Let 1 > 1 2 > 0. According to Lemmas 15 and 18, there exists a measurable part K 1 such that µ 2 (K 1 ) 1-1 , a real M > 0 and a real a > 0, such that

where

Furthermore, we are going to suppose, even if it means making 2 goes to 0, that for every

then L 1 < and the reverse is also true. In this case, according to Theorem 2, one has

where (k 1 , k 2 ) ∈ Π -{(1, 0)}. Thus, by choosing > 0 so that K 2 > , we get that :

when L 1 < . Otherwise, S ,2 (ω, L) = 0 and

1 L 1 < = 0. And so the second result is also true.

The following lemma precise the Lemma 12 and it gives us a better understanding of the law of

. The proof of it is basically the same as the proof of Lemma 12.

Lemma 20. When σ(L)

m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0, there exists C > 0 such that

and this equality is true for every β large enough.

Proof. The proof is a direct application of Lemma 9.

We recall a classic tool from probability theory : Lemma 21. For X a real random variable, one has for every k 1

Thus, by using the fact that Z 1,0 is bounded and is different from 0 and by using Lemma 20 and Lemma 21, we get the following fact about the behaviour

admits a moment of order 1 + κ.

Furthermore,

does not admit a moment of order 4 3 when σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0.

We finally remind the reader of the following obvious lemma : Lemma 23. If X is a real random variable integrable then if it is symmetrical, its expectation is equal to zero.

We can now prove the Proposition 7.

Proof of Proposition 7. Because of the Lemma 23, we only need first to prove that SA (ω, L) converges almost surely when A → ∞ and that the limit random variable is symmetrical and admits moment of order 1 + α and that for all 0 α < 1 3 : the fact that the expectation of the limit is null will be a consequence of Lemma 23.

• Let's prove that SA (ω, L) converges almost surely when A → ∞. It is enough to prove that it is the case on all the events { L 1 } where > 0. So, let's give us > 0. On such an event, one has :

Yet, one has, because the X k are symmetrical and integrable,

By using (109) (see Lemma 19) and the equation ( 119) and because the Z k are independent between them and their expectations are equal to zero, one gets that :

S ,1,A (ω, L) converges almost surely when A → ∞.

Because of equation ( 118), one gets that SA (ω, L) converges almost surely when A → ∞ on all the events { L 1 } where > 0.

• The fact that lim A→∞ SA (ω, L) is symmetrical follows directly from the fact that the Z k are symmetrical.

• It only remains to prove that lim A→∞ SA (ω, L) admits a moment of order 1 + κ and that for all 0 κ < 1 3 ; that lim A→∞ SA (ω, L) does not admit a moment of order 4 3 when σ(L) m where m > 0 and where L belongs to an event of the form ( L < α) with α > 0. ; and that if μ2 is such that there exists α > 0 such that μ2 ({L ∈ S 2 | L 1 < α}) = 0 then the limit lim A→∞ SA (ω, L) admits moments for all orders 1 p < ∞. These facts are a direct consequence of Lemma 19, of Lemma 22 and of Fatou's lemma.


T Wauters
email: tom.wauters@iter.org

D Matveev

D Douai

J Banks

R Buckingham

I S Carvalho

E De La Cal

E Delabie

T Dittmar

J Gaspar

A Huber

I Jepu

J Karhunen

S Knipe

M Maslov

A Meigs

I Monakhov

V S Neverov

C Noble

G Papadopoulos

E Pawelec

S Romanelli

A Shaw

H Sheikh

S Silburn

A Widdowson

P Abreu

S Aleiferis

J Bernardo

D Borodin

S Brezinsek

J Buermans

P Card

P Carvalho

K Crombe

S Dalley

L Dittrich

C Elsmore

M Groth

S Hacquin

R Henriques

V Huber

P Jacquet

X Jiang

G Jones

D Keeling

D Kinna

K Kirov

M Kovari

E Kowalska-Strzeciwilk

A B Kukushkin

H Kumpulainen

E Litherland-Smith

P Lomas

T Loarer

C Lowry

A Manzanares

A Patel

A Peacock

P Petersson

N Petrella

R A Pitts

J Romazanov

M Rubel

P Siren

T Smart

E R Solano

Ž Štancar

J Varje

A Whitehead

S Wiesen

M Zerbini

M Zlobinski

Jet Contributors

Isotope removal experiment in JET-ILW in view of Tremoval after the 2 nd DT campaign at JET

Keywords: Tritium, Inventory, JET, DTE2, ITER, Wall conditioning

A sequence of fuel recovery methods was tested in JET, equipped with the ITER-like beryllium main chamber wall and tungsten divertor, to reduce the plasma deuterium concentration to less than 1% in preparation for operation with tritium. This was also a key activity with regard to refining the clean-up strategy to be implemented at the end of the 2 nd DT campaign in JET (DTE2) and to assess the tools that are envisaged to mitigate the tritium inventory build-up in ITER. The sequence began with 4 days of main chamber baking at 320ºC, followed by a further 4 days in which Ion Cyclotron Wall Conditioning (ICWC) and Glow Discharge Conditioning (GDC) were applied with hydrogen fuelling, still at 320ºC, followed by more ICWC while the vessel cooled gradually from 320ºC to 225ºC on the 4 th day. While baking alone is very efficient at recovering fuel from the main chamber, the ICWC and GDC sessions at 320ºC still removed slightly higher amounts of fuel than found previously in isotopic changeover experiments at 200ºC in JET. Finally, GDC and ICWC are found to have similar removal efficiency per unit of discharge energy. The baking week with ICWC and GDC was followed by plasma discharges to remove deposited fuel from the divertor. Raising the inner divertor strike point up to the uppermost accessible point allowed local heating of the surfaces to at least 800ºC for the duration of this discharge configuration (typically 18 s), according to infra-red thermography measurements. In laboratory thermal desorption measurements, maintaining this temperature level for several minutes depletes thick co-deposit samples of fuel. The fuel removal by 14 diverted plasma discharges is analysed, of which 9, for 160 s in total, with raised inner strike point. The initial D content in these discharges started at the low value of 3-5%, due to the preceding baking and conditioning sequence, and reduced further to 1%, depending on the applied configuration, thus meeting the experimental target.

Introduction

Since the installation of the beryllium (Be) main chamber wall and tungsten (W) divertor, the JET tokamak is uniquely equipped to study material erosion and migration in the vessel along with the retention and removal of fuel in the most ITERrelevant conditions [START_REF] Matthews | Overview of the ITER-like wall project*[END_REF]. Campaign-averaged information on long-term fuel retention and material migration processes is systematically obtained, ex-situ, by post-mortem analysis of components removed from the JET vessel after operation campaigns [START_REF] Likonen | First results and surface analysis strategy for plasma-facing components after JET operation with the ITER-like wall[END_REF]- [START_REF] Widdowson | Overview of fuel inventory in JET with the ITER-like wall[END_REF]. Such analysis after the third ITER-like wall (ILW) campaign (ILW-3, executed in 2015-2016) confirmed the previous trends found for ILW-1 and ILW-2: fuel retention in JET is dominated by co-deposition with Be, for which the thickest deposits, up to a few tens of µm, are found on Tiles 0 and 1 (see section 4) of the inner divertor [START_REF] Widdowson | Evaluation of tritium retention[END_REF]. For the ILW-3 campaign, this location retains 46.5% of all the accumulated fuel while 13.4% is found at the outer divertor, 14.3% on the main chamber limiters and upper dump plates, 10.5% on recessed areas of the main chamber and 12.4% in main chamber gaps such as castellation grooves on plasma-facing components (PFC). The global retention rate for ILW-3, normalised to the divertor plasma time, is 6.3×10 18 D/s [START_REF] Likonen | First results and surface analysis strategy for plasma-facing components after JET operation with the ITER-like wall[END_REF] while gas balance analysis, assessing the fuel inventory in situ and on shorter timescales, estimates 0.2 -1.5×10 20 D/s [START_REF] Heinola | Fuel retention in JET ITER-Like Wall from post-mortem analysis[END_REF], [START_REF] Brezinsek | Fuel retention studies with the ITER-Like Wall in JET[END_REF], [START_REF] Loarer | Plasma isotopic changeover experiments in JET under carbon and ITER-like wall conditions[END_REF] with a spread that is explained by the strong dependence of retention on the discharge scenarios and the chosen outgassing duration. The same effect, namely the dependence of the dynamic and long-term retention reservoirs on the considered outgassing time after plasma operation [START_REF] Philipps | Dynamic fuel retention and release under ITER like wall conditions in JET[END_REF], explains the difference between the results of ex-situ and in-situ retention analysis at JET. Fuel recovery techniques, which are the topic of this paper, target the long-term retention. Tokamaks working with tritium (T) may have several reasons to apply such techniques. They are essential to the clean-up strategy to be implemented at the end of the 2 nd DT campaign in JET (DTE2). Moreover, the ongoing JET campaigns, operating with deuterium (D), T and D-T mixtures have strict limits on the number of 14 MeV neutrons produced by fusion reactions in each of the campaign phases. These are set so as not to unnecessarily consume the total neutron budget of the facility set by vessel activation limits [START_REF] Joffrin | Overview of the JET preparation for deuterium-tritium operation with the ITER like-wall[END_REF]. This requires that the residual D content in the plasma be reduced to 1% from the start of the T-phase, and similarly for the residual T content in the D-phase following D-T operation. The JET DT safety case limits the potentially releasable T inventory (by loss of vacuum) at any given time to 15 g [START_REF] Boyer | JET Tokamak, preparation of a safety case for tritium operations[END_REF], and assumes that the long-term T retention is at most 4 g. In ITER, the maximum allowed retention in the in-vessel components is set at 700 g. At a JET-like retention fraction of the injected fuel of 0.14-0.19% [START_REF] Widdowson | Evaluation of tritium retention[END_REF], without recovery methods, ITER could potentially reach this limit already within the first fusion power operation (FPO) campaigns [START_REF] Joffrin | Overview of the JET preparation for deuterium-tritium operation with the ITER like-wall[END_REF]. The actual retention fraction is sensitively dependent on wall surface temperatures and incident particle fluences/energies, which may be very different in ITER [START_REF] Pitts | Progress in the understanding of plasma-wall interactions in support of the ITER Research Plan[END_REF]. For these reasons it is essential that the fuel recovery techniques available to JET and ITER be tested and characterized as thoroughly as possible.

The fuel outgassing efficiency from PFCs exposed in JET-ILW has been studied at ITER-relevant baking temperatures, suggesting that fuel removal from thick deposits will be difficult [START_REF] Heinola | Long-term fuel retention and release in JET ITER-Like Wall at ITER-relevant baking temperatures[END_REF], [START_REF] De Temmerman | Efficiency of thermal outgassing for tritium retention measurement and removal in ITER[END_REF]. Only 13% of fuel could be removed from 40 µm thick co-deposit layers on the W coated CFC divertor tiles by 15 hours of baking at 350°C [START_REF] Heinola | Long-term fuel retention and release in JET ITER-Like Wall at ITER-relevant baking temperatures[END_REF]. The JET ILW-1 campaign deposited such layers across the entire campaign, whereas this may occur in only a few days of ITER operation under Q = 10, burning plasma conditions [START_REF] De Temmerman | Efficiency of thermal outgassing for tritium retention measurement and removal in ITER[END_REF]. Removal of implanted fuel from the JET Be main chamber samples in the deposit-free erosion zones of limiter tiles on the midplane by baking at 240°C was even less efficient [START_REF] Heinola | Long-term fuel retention and release in JET ITER-Like Wall at ITER-relevant baking temperatures[END_REF]. From these and similar studies, it is known that all fuel (whether or not in co-deposited layers) is recovered when the surfaces are heated to temperatures of at least 800°C for several minutes [START_REF] Heinola | Long-term fuel retention and release in JET ITER-Like Wall at ITER-relevant baking temperatures[END_REF]- [START_REF] Založnik | Deuterium removal from beryllium co-deposits by simulated strike-point sweeping[END_REF].

The ITER wall conditioning and fuel removal techniques include baking, glow discharge cleaning (GDC) and RF discharges without plasma current, both in the ion cyclotron and electron cyclotron range of frequencies (ICWC and ECWC). JET does not have an electron cyclotron heating system, but routinely uses limiter and diverted plasmas with nested magnetic flux surfaces for cleaning purposes. Moreover, its flexible configuration allows it to place the inner divertor strike point on top of Tile 1, high up on the inner vertical target, so that thermal plasma can be used to heat the local deposits.

Experiment strategy

The JET experiment presented here combines the available fuel removal techniques to bring the D concentration in hydrogen (H) plasma down to 1% before the start of the T operations campaign. The rationale behind the experimental strategy is that fuel removal is accelerated at higher temperatures, but that baking alone, at 320°C for the first wall (FW) and about 200°C for the upper part of the lower divertor, is insufficient to exhaust the inventory completely. Figure 1 shows the timeline of the experiment. One day after the end of the D operations phase, the divertor cryopumps were warmed-up to liquid nitrogen temperature and the vessel temperature was ramped from the 200°C temperature used for normal operations, up to 320°C at a rate of about 5°C per hour. During this time, the divertor maintains the cooling water flow at room temperature to protect the in-vessel divertor coils. This results in a temperature gradient over the divertor tiles, where Tiles 0 and 1 are at most at 150-200°C, the vertical tiles 3 and 7 at 100-150°C and the bottom tiles 4 and 6 at least at 50°C, based on thermocouple trends. The locations of these tiles are indicated in Fig. 3c. The steady baking temperature is maintained for 7 days, in the second half of which, cycles of ICWC and GDC in H are performed. Two days after completing the baking procedure, the cryopanels were cooled to liquid helium temperature in preparation for diverted cleaning plasmas with raised inner strike points, aiming at heating the Tile 1 surface layers.

Sub-divertor residual and optical gas analysers (RGA and OGA) [START_REF] Jepu | Subdivertor RGA and OGAs[END_REF], [START_REF] Kruezi | Neutral gas analysis for JET DT operation[END_REF] are used to quantify the isotopic content in the outgassed molecules. The data is compared to mass spectrometry data from the midplane pumping ducts, H α high-resolution spectroscopy of the main chamber and divertor [START_REF] Neverov | Determination of isotope ratio in the divertor of JET-ILW by high-resolution Hα spectroscopy: H-D experiment and implications for D-T experiment[END_REF], as well as the quantifications and chromatography of gas collected by the pumps [START_REF] Romanelli | Upgraded Analytical Gas Composition Technique in the Tritium Fuel Cycle of JET[END_REF]. The temperatures of the plasma-facing surfaces are monitored by infra-red (IR) thermography [START_REF] Balboa | Recent developments of in-vessel calibration of mid-IR cameras at JET[END_REF] and subsurface thermocouples [START_REF] Guillemaut | Main chamber wall plasma loads in JET-ITER-like wall at high radiated fraction[END_REF].

Baking at 320°C with ICWC and GDC

Figure 1: Timeline of the isotopic exchange experiment indicating the baking duration, the ICWC, GDC and diverted plasma operation sessions, the cooling of the divertor cryopanels, either by liquid helium or nitrogen, the collection of pumped gas at the Active Gas Handling System (AGHS) and reference pulses in the V5 divertor configuration. "Early" and "Late" refer to the two operational shifts which JET runs on each experimental day.

Figure 2 shows the vessel temperature and the partial pressures of HD and D 2 taken in the midplane pumping ducts during the baking week. As the temperature of the vessel rises, thermal desorption of D tracks the temperature waveform, as can be seen in the RGA traces. During the first 24 hours following the achievement of the 320°C plateau, the outgassing pressure decays by a factor of ~3. Beyond this point, it decreases more slowly, typical of the often-observed power law outgassing dependency [START_REF] Philipps | Dynamic fuel retention and release under ITER like wall conditions in JET[END_REF]. The discontinuities in the pressure traces at time intervals of 6 hours are due to background measurements during which the RGAs are temporarily isolated from the torus. A jump in the D 2 pressure is visible at 3.6 days, corresponding to a slow leak through gas injectors that have been prepared to perform gas puffs for RGA and OGA calibrations. These calibrations took place around day 5 on the graph, and the associated pressure data has been removed for clarity.

Hydrogen ICWC was applied in two phases, 27 pulses before GDC and 25 pulses after GDC, as also seen in Fig. 2. The pressure peaks are the result of the post-discharge release of gas from the PFCs, while the steady discharge pressure is controlled at 1.8 × 10 -5 mbar. These ICWC discharges, targeting the replacement of near surface retained D atoms by isotopic exchange with H, were produced by coupling 100-250 kW of power in the ion cyclotron range of frequencies (ICRF), at 29 MHz, in the presence of a toroidal field in most cases of 1.9 T and a barrel shaped vertical field of 15 mT on axis [START_REF] Wauters | Wall conditioning in fusion devices with superconducting coils[END_REF]. A tangential camera view of an ICWC plasma in JET is shown in Fig. 3a to emphasize the uniformity of the RF glow with apparent strongest interaction at the inboard and outboard limiters. The experiment applied mostly 17 s long ICWC pulses, repeated every 15-20 minutes to allow for diagnostic post-processing and machine preparations for these JET pulse-based discharges (required because the toroidal field is not permanently on in JET). It is important to note that ITER (where the toroidal field will be permanently energized during operational campaigns) will apply RF conditioning programmes as sequences of multiple discharges, similar to the so-called pulse trains in W7-X [START_REF] Goriaev | Wall conditioning at the Wendelstein 7-X stellarator operating with a graphite divertor[END_REF]. This will allow higher repetition rates and duty cycle optimisation to minimise the re-deposition of wall released isotopes.

The GDC on JET in the experiments reported here was performed with 3 anodes at 2.5-4 A and 205-275 V, delivering 2.4 kW of DC power to the plasma, without magnetic field, at a steady neutral pressure of 3.1 × 10 -3 mbar. A camera view of a GDC plasma is shown in Fig. 3b, illustrating the uniformity of the discharge and the anode location at the top of the vessel where the bright anode glow is seen. Hydrogen GDC was applied in two continuous procedures of ~17 hours and ~16 hours respectively. A 1 hour-long discontinuity is visible in Fig. 2 about halfway through each glow phase when the background neutral gas at the midplane RGA was monitored. The glows are separated by 2 hours, as seen in the outgassing traces of Fig. 2 starting at 7.7 days, to provide better understanding of the D removal process. The first glow displays a rapid decrease of the outgassing by a factor of 3 in the first 6 hours, with thereafter a slower trend, typical of the power law outgassing dependency referred to earlier. The second glow does not show this feature and continues the outgassing rate approximately at the level where it was interrupted. Thus, the accessible isotope reservoir does not seem to be significantly repopulated by diffusion from deeper layers in the dwell time between the two glows, even at baking temperatures. Therefore, the accessible reservoir may be rather determined by the extent of the transient loading of the surfaces by isotopes provided via the conditioning discharge and the mechanism of isotopic exchange. Since the JET GDC system does not allow the glow to be initiated at operating pressure, unlike in e.g. ASDEX Upgrade [START_REF] Härtl | Optimization of the ASDEX Upgrade glow discharge[END_REF], it is difficult to investigate this further in JET, nor is it possible to implement a more meaningful discharge cycle which minimises re-deposition or material sputtering.

Gas balance

Baking, from the start of the temperature rise to the first ICWC discharge, corresponding to 5.4 days in Fig. 2, removed 2.9 × 10 23 D atoms from the vessel. This amount is an average of partial pressure measurements with 2 midplane and 4 subdivertor RGAs, and estimated extrapolation of the pressures along the outgassing trend described by the dashed lines in Fig. 2. Extrapolating the trend to the last ICWC discharge at the end of the baking week, corresponding to 9 days in Fig. 2, predicts that an additional 1.2 × 10 23 D atoms could have been removed by continued baking without ICWC or GDC. This latter number is probably overestimated and should be used with caution given the uncertainty of the extrapolation.

The removal in the ICWC sessions is obtained by collecting and analysing all the pumped gas, starting from the first ICWC discharge and ending with 1.5 hours of outgassing after the last ICWC discharge in each of the two ICWC phases. The first phase (Sessions 3-5 in Fig. 1) with an energy throughput of 85.7 MJ performed over 27.9 hours removed 7.3 × 10 22 D atoms while the second (Sessions 9-10 in Fig. 1) removed 1.7 × 10 22 D for an energy throughput of 57.4 MJ over 13.9 hours (recall that the ICWC is only applied in short pulses separated by 15-20 minute intervals and that the usual outgassing occurs between each set of pulses). For completeness, the removed amount estimated from 2 sub-divertor RGAs is given, each combining the pulse-based recordings (not available for the midplane RGAs) with a time resolution of 0.33 seconds per sample and per mass channel, and the continuous data measured in-between pulses at a lower time resolution. Since deconvolution of this RGA data becomes increasingly unreliable at decreasing isotopic concentration, the removal is underestimated by a factor of ~2.

Similarly, since the deconvolution of the sub-divertor RGAs became unreliable in the last 16 hours of the procedure with increasingly lower pressures, the gas balance for the GDC sessions is obtained from the 2 midplane RGAs only. An estimated 1.3 × 10 23 D atoms were removed by the 33 hours of GDC, corresponding to a total energy input of 285 MJ.

Applying GDC and ICWC in addition to baking yields a two-fold gain compared to baking alone. ICWC applied after an extensive GDC procedure still removes a significant quantity of D atoms, but less than the first ICWC sessions. From this it is concluded that both GDC and ICWC share to a large extent their plasma-wall interaction areas.

ICWC and GDC may be considered to have at best a uniform wetting of the plasma-facing surfaces (see Fig. 3 a andb). The first field lines intersecting components during ICWC plasmas, such as main chamber limiters, carry most of the ion flux, while the recessed areas including gaps such as castellations and recessed areas on the main wall are conditioned by a flux of neutral atoms [START_REF] Tabarés | Plasma-Assisted Wall Conditioning of Fusion Devices: A Review[END_REF], [START_REF] Douai | Ion cyclotron wall conditioning in KSTAR and ASDEX-upgrade[END_REF]. Gas balance analysis, which includes the removal trends discussed in the next sections, integrates the removal from all these areas. In-situ diagnostic techniques, such as Laser Induced Desorption with detection by Quadrupole Mass Spectrometry (LID-QMS), currently being prepared for JET [START_REF] Zlobinski | [END_REF] and ITER, are required to assess the removal from a particular location. The removed amounts throughout the baking week, estimated by gas balance analysis, are a significant fraction of the inventory based on post-mortem analysis (43%), as detailed in [START_REF] Widdowson | Evaluation of tritium retention[END_REF]. However, the analysis of the extracted material samples is completed 1-2 years after their last plasma exposure in JET, while outgassing continues throughout this period. Similarly, only local, in-situ measurements could provide some estimate of the inventory at the start of this experiment, built up during plasma exposure and slowly, but continuously depleted by outgassing during the campaign period (running over 1-2 years).

Gain of combining ICWC and GDC with baking

Earlier work reported a removal of 10 23 atoms by 1.26 hours of GDC at 200°C in JET-ILW [START_REF] Douai | Wall conditioning for ITER: Current experimental and modeling activities[END_REF]. The same duration in the present experiment at 320°C removed 1.2 × 10 22 D atoms (where the 1.26 hours is counted from the establishment of the GDC discharge). This almost an order of magnitude difference is due to the depletion of the surface of D by the several days of baking and the ICWC sessions which preceded the GDC. This clearly illustrates that substantially more fuel can be recovered by extending the conditioning procedure; the accessible reservoir depends on the conditioning duration. Thus, although the first ~1.5 hours of GDC in the current experiment at 320°C removed less than the older reported result, the total removal over the 33 glow hours here did exceed that obtained at 200°C despite the preceding surface depletion.

In addition to duration, the accessible reservoir for isotope exchange depends on the surface temperature. Comparison of the cumulated removal by ICWC capped at an energy throughput of 85 MJ in the first ICWC phase before GDC, and previous ICWC work at 200°C in JET-ILW [START_REF] Wauters | Isotope exchange by Ion Cyclotron Wall Conditioning on JET[END_REF], gives 7.3 × 10 22 and 5.6 × 10 22 atoms, respectively. Thus, ICWC at 320°C appears to be more effective than at normal JET operating temperature, even despite the prior baking days. The same energy throughput in GDC removed 6.2 × 10 22 atoms, only slightly lower, despite the preceding desaturation by ICWC.

The vessel temperature was ramped down from 320°C to 225°C during the last ICWC phase (Sessions 9-10 in Fig. 1). While the D content is fairly constant at first in the ICWC discharges, still at constant vessel temperature, it steadily decreases with the decreasing temperature, as shown in Fig. 4. This again indicates a reduced removal efficiency at a lower surface temperature, which is of particular concern for ITER, where ICWC will be used while the PFCs are cooled to 70°C (the ITER ion cyclotron resonance heating (ICRH) antennas cannot be used at FW baking temperature). Dedicated isotope exchange experiments which investigate the temperature dependence are therefore required, both in JET, where ICWC may be operated at 110°C, approaching the ITER-relevant surface temperature, and in dedicated facilities such as TOMAS [START_REF] Goriaev | The upgraded TOMAS device: A toroidal plasma facility for wall conditioning, plasma production, and plasmasurface interaction studies[END_REF] where samples can be exposed to ICWC and GDC at controlled temperatures. power law envelope curve with exponent for the integrated removal. A curve with this same exponent has been added (dashed line) to illustrate the similarity between the ICWC and GDC energy efficiency, expressing the total removal as a function of the cumulated energy throughput. Here it is considered that the input power is a measure for the ion flux in GDC [START_REF] Kogut | Modelling of tokamak glow discharge cleaning II: comparison with experiment and application to ITER[END_REF], and neutral flux in ICWC [START_REF] Moon | Characterization of neutral particle fluxes from ICWC and ECWC plasmas in the TOMAS facility[END_REF], to the main chamber surfaces. It is important to note that the trend (dashed) line shows no saturation even though the integral of the coupled ICRF and GDC power exceeds the previous experiments by a factor of 3. This indicates that significantly more atoms may still be removed by extending the conditioning procedure, which is typical for the power law dependency. By extending the procedure, the accessible reservoir increases due to an increased discharge energy throughput.

Total removal and discharge energy throughput

If the power throughput is indeed the rate determining factor, the energy consumption of GDC and ICWC needs to be mapped to the wall-clock time. In a tokamak with superconducting coils (such as ITER, but unlike JET), the equivalent discharge energy of 33 hours of GDC at 2.4 kW can be applied using ICWC at 250 kW of coupled power within a period of 3 hours at a 10% duty cycle. This means that fuel removal by ICWC may be 10 times faster than GDC. While this number is striking, the duty cycle must be considered in more detail, which may be performed in silico by reaction diffusion modelling of isotopic exchange. Indeed, outgassing from PFCs is a slow process that continues during the dwell time between the ICWC pulses. Post-mortem analysis of JET PFCs showed that 46.5% of the retained fuel is deposited along with Be on the upper part of the inner divertor [START_REF] Widdowson | Evaluation of tritium retention[END_REF]. In order to deplete these surface layers from fuel by thermal outgassing alone, they must be heated above 800 ºC for several minutes, based on TDS spectra [START_REF] Založnik | Deuterium removal from beryllium co-deposits by simulated strike-point sweeping[END_REF]. To achieve these temperature conditions locally, in situ, a diverted plasma scenario was developed with a stationary phase of 18 s in which the inner strike point is raised up to the uppermost accessible point on Tile 1, where the thickest deposits reside. The resulting raised inner strike point (RISP) configuration is illustrated by the blue flux surfaces in Fig. 3c. For comparison, the green flux surfaces show the typical JET V5 configuration. Following the week of high temperature baking, a total of 9, main chamber hydrogen fuelled RISP discharges have been performed at the usual 200°C vessel operating temperature. These L-mode discharges at 1.8 T, 1.7 MA and 2 to 4.5 MW of ICRH power at 51 MHz reached a central line averaged density of 2.3 × 10 19 m -3 , with high recycling conditions at the inner strike point characterised by a peak temperature of 5-10 eV and density of 1.5 × 10 20 m -3 at the target.

Diverted plasma with raised inner strike point

In total 2.4 × 10 22 D atoms have been removed in 14 diverted plasma discharges, including the 9 RISP pulses, as obtained from volumetric and chromatography analysis of the pumped gas. The removed amount represents 12.5% of the long term retention at Tiles 0 and 1 found in post-mortem analysis. Comparing these numbers ignores, however (i) the possible depletion of the divertor deposits during the baking week with ICWC and GDC, (ii) the removal from other areas during the diverted discharges, (iii) the re-deposition of fuel removed from Tiles 0 and 1 during the discharge and (iv) the continuous slow outgassing of these layers before their ex situ analysis as described in Section 3. This again emphasises the importance of in situ quantification of the fuel content in the divertor PFCs.

The limited thermal conductivity of the deposits to the substrate layer helps in heating them during the plasma exposure [START_REF] Gaspar | In-situ estimation of the thermal resistance of carbon deposits in the JET tokamak[END_REF]. Infra-red images of the divertor (Fig. 6) show that the enhanced heat flux brought the Tile 1 surface temperature to the target value of 800°C and above, for the duration of the discharges. At the same time, it is observed that the heating is nonuniform, both in the poloidal and toroidal directions. The sharp temperature variations seen at the end of the discharge and during a strike point sweep applied around 60 s in Fig. 6b, are consistent with the assumption of resistive surface deposits. Thermocouple data (not shown here) indicate only a small ratcheting of the tile base temperature of about 50°C throughout the subsequent RISP pulses. While heating of the bulk material of about 100°C per pulse was observed, reaching a maximum temperature of about 200°C overall, the pulse repetition rate was clearly too low to substantially increase the base temperature.

RISP reaches high Tile 1 surface temperatures with modest input power. It appears to be challenging to reach such high surface temperatures with other plasma configurations. The standard V5 configuration (green flux surfaces in Fig. 3c) attains 500°C at Tile 1 thanks to a relatively small distance between the inner baffle and the separatrix (interestingly, in the present experiment, a Tile 1 surface temperature of about 450°C was reached in ICWC discharges where the fundamental resonance layer of the D minority was placed at the inner divertor). While these temperatures are insufficient for complete thermal outgassing of the layers, these vertical plasma configurations (which are the standard configurations used on ITER) may be considered as a means to limit the fuel content in the inner divertor layers through isotopic exchange, before subsequent deposits build up. The feasibility of plasma-induced removal from deposits as a potential means of T control in ITER was explored in [START_REF] Založnik | Deuterium removal from beryllium co-deposits by simulated strike-point sweeping[END_REF], and needs to be investigated further, though the active cooling of PFCs in ITER may make it difficult to raise the surface temperatures significantly, especially far from the strike points where most of the deposits are expected to accumulate, unless their conductivity to the W divertor monoblocks is sufficiently low. Figure 7 summarises the isotopic content in the diverted plasmas measured by the sub-divertor OGA and spatiallyresolved H α high-resolution spectroscopy. As a consequence of the preceding conditioning sequence, described in Section 3, the D content in a reference discharge with V5 configuration started at the low value of 3-5% in plasma. It subsequently decreased further to 1%, depending on the applied configuration, in the 14 discharges (including the 9 RISP pulses). This then meets the overall target of these inventory control experiments.

The spectroscopy measurements at the upper inner divertor (KSRD [START_REF] Neverov | Determination of isotope ratio in the divertor of JET-ILW by high-resolution Hα spectroscopy: H-D experiment and implications for D-T experiment[END_REF] channels i1-i3 in Fig. 7) consistently show a higher D content in the RISP discharges compared to other divertor locations, indicative of the locally stimulated thermal outgassing. While the difference between these measurements decreases from pulse to pulse, it can be seen that the changeover on Tile 1 is not yet complete on the last RISP pulse.

The isotopic content of the neutral gas in the sub-divertor, representative of the pumped gases, is consistently higher than the divertor spectroscopy measurements in non-RISP discharges, indicating that additional D fuel is released in these discharges from locations not accessible by RISP. Separately, it is recognised that the sensitivity limit for isotopic ratio measurements by the OGA is close to the resulting D content in H of 1-2%.

Conclusion

A sequence of fuel recovery schemes following a long deuterium plasma operation campaign was tested in JET to reduce the plasma D concentration to below 1%. The experiment aimed at refining the clean-up strategy to be implemented at the end of the JET DTE2 campaign and assessed tools that are envisaged to mitigate the tritium inventory build-up in ITER. The sequence in JET, equipped with the ITER-like beryllium main chamber wall and tungsten divertor, began with 4 days of baking removing 2.9 × 10 23 D atoms, presumably from the main chamber surfaces. While the vacuum vessel is heated to 320ºC in this procedure, the JET divertor continues to be cooled to protect in-vessel components, keeping the upper divertor surfaces below 150-200ºC and still lower for surfaces lower down in the divertor. In the subsequent 4 days, (i) Ion Cyclotron Wall Conditioning (ICWC) and (ii) Glow Discharge Conditioning (GDC) were applied with hydrogen fuelling, still at 320ºC, followed by (iii) hydrogen ICWC while the vessel cooled gradually from 320ºC to 225ºC on the 4 th day. Gas balance analysis yields estimates of the removal at respectively (i) 7.3 × 10 22 , (ii) 1.3 × 10 23 and (iii) 1.7 × 10 22 D atoms. While baking alone removed a significant amount of fuel, the subsequent ICWC and GDC sessions still removed slightly higher amounts than found previously in isotopic changeover experiments in JET with the vacuum vessel at 200ºC.

The GDC procedure, consisting of two glows of 17 and 16 hours, separated by 2 hours for passive outgassing, found no evidence of temporarily increased D outgassing in the second GDC. The isotope flux from the conditioning plasma, rather than diffusion in the material, seems to be the process determining factor at the timescale of this experiment. The removal by GDC as a function of the coupled energy (285 MJ in total), continues approximately along the same power law fit for the removal by ICWC (85.7 MJ in total before and 57.4 MJ after GDC), indicating a comparable particle flow per unit of coupled energy and equivalent plasma-material interaction areas. A clear positive correlation between fuel removal and surface temperature is found in the ICWC pulses at different main chamber wall temperatures during the vessel cool down from 320ºC to 225ºC.

About half of the trapped fuel is co-deposited on the upper part of the lower divertor inner target, as evidenced by postmortem analysis following previous JET-ILW campaigns [START_REF] Widdowson | Evaluation of tritium retention[END_REF]. Raising the inner divertor strike point onto Tile 1 in a specially designed scenario allowed these co-deposited layers to be locally heated above 800ºC for the duration of the tokamak discharge. In total, 2.4 × 10 22 D atoms were removed in 14 discharges, 9 of which (for an integral duration of 162 s) with raised inner strike point. The plasma D content in these discharges started at the low value of 3-5%, due to the preceding baking and conditioning sequence, and reduced further to 1% depending on the applied plasma configuration, thus meeting the overall target set for this fuel recovery experiment.

The amounts removed in this experiment approach the long-term inventory obtained by post-mortem analysis of material samples extracted from the JET PFCs (43% of the retention measured in [START_REF] Widdowson | Evaluation of tritium retention[END_REF]). To make this comparison quantitative requires a correction for the continuous slow outgassing of these samples before they are analysed ex situ. Since such a scaling over a long time period is inevitably inaccurate, in situ quantification of the in-vessel inventory content is very much needed. The V5 and RISP divertor configurations are illustrated in Fig. 3c. The VT and CC configurations put the inner and outer strike points on the vertical targets (Tiles 3 and 7 in Fig. 3c) and lower corner tiles (Tiles 4 and 6 in Fig. 3c) respectively.

Figure 2 :

 2 Figure 2: Partial pressures of HD and D2 (left axis) and vessel temperature (right axis) throughout the baking week.

Figure 3 :

 3 Figure 3: Tangential camera images of typical ICWC (a) and GDC (b) discharges in JET. Right (c): Magnetic flux surfaces at the divertor for the Raised Inner Strike Point (RISP) and V5 configuration with numbering of the divertor tiles in red.ICWC GDC

Figure 4 :

 4 Figure 4: D content from sub divertor OGA in ICWC discharges of Sessions 9-10 (Fig. 1) and vessel temperature.

Figure 5 ,

 5 Figure5, adapted from[START_REF] Wauters | Wall conditioning in fusion devices with superconducting coils[END_REF], shows cumulated removal tendencies for earlier ICWC experiments in JET-ILW, with the present ICWC (purple asterisk) and GDC (continuous green line) experiment added. Plotted as function of the cumulated coupled ICRF energy, these experiments followed a power law envelope curve with exponent for the integrated removal. A curve with this same exponent has been added (dashed line) to illustrate the similarity between the ICWC and GDC energy efficiency, expressing the total removal as a function of the cumulated energy throughput. Here it is considered that the input power is a measure for the ion flux in GDC[START_REF] Kogut | Modelling of tokamak glow discharge cleaning II: comparison with experiment and application to ITER[END_REF], and neutral flux in ICWC[START_REF] Moon | Characterization of neutral particle fluxes from ICWC and ECWC plasmas in the TOMAS facility[END_REF], to the main chamber surfaces. It is important to note that the trend (dashed) line shows no saturation even though the integral of the coupled ICRF and GDC power exceeds the previous experiments by a factor of 3. This indicates that significantly more atoms may still be removed by extending the conditioning procedure, which is typical for the power law dependency. By extending the procedure, the accessible reservoir increases due to an increased discharge energy throughput.If the power throughput is indeed the rate determining factor, the energy consumption of GDC and ICWC needs to be mapped to the wall-clock time. In a tokamak with superconducting coils (such as ITER, but unlike JET), the equivalent discharge energy of 33 hours of GDC at 2.4 kW can be applied using ICWC at 250 kW of coupled power within a period of 3 hours at a 10% duty cycle. This means that fuel removal by ICWC may be 10 times faster than GDC. While this number is striking, the duty cycle must be considered in more detail, which may be performed in silico by reaction diffusion modelling of isotopic exchange. Indeed, outgassing from PFCs is a slow process that continues during the dwell time between the ICWC pulses.

Figure 5 :

 5 Figure 5: Total removal vs. discharge energy throughput for earlier ICWC experiments on JET-ILW at 25MHz (II, III & IV [21]), and the present combined ICWC (VI) and GDC experiment. The dashed line is a possible envelope curve for the total removal in case of a previously observed E 0.5 removal dependency [21]

Figure 6 :

 6 Figure 6: Divertor IR thermography illustrating the geometric (a) and temporal (b, 1D taken along black curve in a) pattern of the heating at the inner divertor, for a discharge with RISP (JPN98298, camera KL7-E8WB).

Figure 7 :

 7 Figure 7: Isotopic content in diverted plasmas by sub-divertor OGA and spatially-resolved high-resolution H α spectroscopy.The V5 and RISP divertor configurations are illustrated in Fig.3c. The VT and CC configurations put the inner and outer strike points on the vertical targets (Tiles 3 and 7 in Fig.3c) and lower corner tiles (Tiles 4 and 6 in Fig.3c) respectively.

Acknowledgements

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission, nor of the ITER Organization.