Zolt Án

S T Éphane Seuret

Zoltán Buczolich

Stéphane Seuret
email: seuret@u-pec.fr

MULTIFRACTAL PROPERTIES OF TYPICAL CONVEX FUNCTIONS

We study the singularity (multifractal) spectrum of continuous convex functions defined on [0, 1] d . Let E f (h) be the set of points at which f has a pointwise exponent equal to h. We first obtain general upper bounds for the Hausdorff dimension of these sets E f (h), for all convex functions f and all h ≥ 0. We prove that for typical/generic (in the sense of Baire) continuous convex functions f :

, and in addition, we obtain that the set E f (h) is empty if h ∈ (0, 1) ∪ (1, +∞). Also, when f is typical, the boundary of [0, 1] d belongs to E f (0).

Introduction and main results

In this paper we investigate the multifractal properties of the continuous convex functions defined on [0, 1] d . This paper is a quite natural continuation of our papers [START_REF] Buczolich | Typical borel measures on [0, 1] d satisfy a multifractal formalism[END_REF][START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF] where generic multifractal properties of measures and of functions monotone increasing in several variables (in short: MISV) were studied. It is interesting that all these natural objects supported on [0, 1] d have very different typical multifractal behaviors.

Let us first recall that the pointwise Hölder exponent and the singularity spectrum for a locally bounded function are defined as follows.

Definition 1. Let f ∈ L ∞ ([0, 1] d ). For h ≥ 0 and x ∈ [0, 1] d , the function f belongs to C h (x) if there are a polynomial P of degree strictly less than [h] and a constant C > 0 such that, for all x close to x, [START_REF] Bayart | Multifractal spectra of typical and prevalent measures[END_REF] |f (x ) -P (x -x)| ≤ C|x -x| h .

The pointwise Hölder exponent of f at x is

h f (x) = sup{h ≥ 0 : f ∈ C h (x)}.
In the following, dim H denotes the Hausdorff dimension.

Definition 2. The singularity spectrum of f is the mapping

d f (h) = dim H E f (h), where E f (h) = {x : h f (x) = h}.
By convention dim ∅ = -∞. We will also use the sets

(2)

E ≤ f (h) = {x : h f (x) ≤ h} ⊃ E f (h).
We denote by CC d the set of continuous convex functions f : [0, 1] d → R. Equipped with the supremum norm • , CC d is a separable complete metric space. An open ball in CC d of center f ∈ CC d of radius r ≥ 0 is written as B • (f, r), and a closed ball is B • (f, r)

In this paper, we first prove an upper bound for the multifractal spectrum of all functions in CC d . Theorem 1. For any function f ∈ CC d , one has

d f (h) ≤      d -1 if h ∈ [0, 1) d + h -2 if h ∈ [1, 2] 2 if h > 2.
Then we compute the multifractal spectrum of typical functions in CC d . Recall that a property is typical, or generic in a complete metric space E, when it holds on a residual set, i.e. a set with a complement of first Baire category (a set is of first Baire category if it is the union of countably many nowhere dense sets).

Theorem 2. For typical functions f ∈ CC d , one has

d f (h) =      d -1 if h = 0 d + h -2 if h ∈ [1, 2] -∞ otherwise.
More precisely, one has

E f (0) = ∂([0, 1] d ).
It is interesting to compare Theorem 2 with the regularity of other typical objects naturally defined on the cube [0, 1] d .

When d = 1, generic properties of continuous functions on [0, 1] have been studied for a long time (see for instance [START_REF] Gruber | Die meisten konvexen körper sind glatt, aber nicht zu glatt[END_REF][START_REF] Howe | Most convex functions are smooth[END_REF], and the other references we mention in the present paper). It was proved that typical continuous functions belong to C 0 (x), for every x ∈ [0, 1]. In [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF], typical monotone continuous functions on the interval [0, 1] were

f : [0, 1] → R satisfy (3) d f (h) = dim E ≤ f (h) = h if h ∈ [0, 1] -∞ otherwise.
The same holds true for typical monotone functions (not necessarily continuous). We remark that it also follows, for example from results in [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF], that for arbitrary monotone functions there is an upper estimate [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF] dim

E ≤ f (h) ≤ h for h ∈ [0, 1].
It is interesting to extend these results to higher dimensions. The first natural way is to consider Borel measures on the cube. In [START_REF] Buczolich | Typical borel measures on [0, 1] d satisfy a multifractal formalism[END_REF] (see also [START_REF] Bayart | Multifractal spectra of typical and prevalent measures[END_REF] for a nice generalization to all compact sets in R d ), it is proved that typical measures µ supported on [0, 1] d satisfy a multifractal formalism, and

d µ (h) = h if h ∈ [0, d] -∞ otherwise.
Another interesting class is constituted of the continuous monotone increasing in several variables (in short: MISV) functions. These functions extend to higher dimensions in a different direction the onedimensional monotone functions. A function f : [0, 1] d → R is MISV when for all i ∈ {1, ..., d}, the functions ( 5)

f (i) (t) = f (x 1 , ..., x i-1 , t, x i+1 , ..., x d )
are continuous monotone increasing. We use the notation

M d = {f ∈ C([0, 1] d ) : f MISV}.
The space M d is a separable complete metric space when equipped with the supremum L ∞ norm for functions. Typical MISV functions satisfy (see [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF][START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF])

(6) d f (h) = d -1 + h if h ∈ [0, 1] -∞ otherwise.
On Figure 1 we compare our new results about generic continuous convex functions with the earlier results we mentioned.

Remark 1. One cannot directly infer Theorems 1 and 2 by integrating MISV functions or measures on

[0, 1] d . For instance, letting f (x 1 , x 2 ) = 10(x 2 1 + x 2 2 ) + x 2 1 x 2 2 , its second differential d 2 f (x 1 , x 2 , h 1 , h 2 ) = (20 + 2x 2 2 )h 2 1 + (20 + 2x 2 1 )h 2 2 + 4x 1 x 2 h 1 h 2 is positive definite for any (x 1 , x 2 ) ∈ [-1, 1] 2 . Hence this function f is strictly convex on [-1, 1] 2 , but ∂ 1 f = 20x 1 +2x 1 x 2 2 is monotone only in x 1 and ∂ 2 f = 20x 2 +2x 2 x 2 1 is monotone only in x 2 .

Preliminary Results

2.1.

Basic notation. If it is not stated otherwise we work in R d . Points in the space are denoted by x = (x 1 , ..., x d ). The j'th unitvector is denoted by e j = (0, ..., 0, 1

↑ j , 0, ..., 0).
Open balls in R d are denoted by B(x, r) (to be distinguished from the balls B • (f, ε) in CC d ).

Local regularity results of continuous convex functions.

First, recall that C ∞ functions are dense in CC d . Remark 2. To see this, take ψ ≥ 0 a C ∞ function, which is 0 outside B(0, 1), such that

R d ψ = 1. Put ψ λ = λ d ψ( 1 λ x). If f is convex and continuous on [-1, 1] d , then the convolution f λ (x) = [0,1] d f (x)ψ λ (y - x)dy is convex on [-1 + λ, 1 -λ] d and f λ (x) = f λ ( 1 1-λ x) is convex on [-1, 1] d .
Using the uniform continuity of f on [-1, 1] d , one easily sees that ||f -f λ || → 0 as λ → 0. One concludes by using an obvious linear transformation.

A first lemma allows one to control the left and right partial derivatives of all functions in a neighborhood of a convex differentiable function in CC d . The notations ∂ j,+ f and ∂ j,-f are used for the right and left partial j-th derivatives of a convex function f . Lemma 3. Suppose f ∈ CC d ∩ C 1 ([0, 1] d ) and ε > 0. There exists f,ε > 0, such that for all j ∈ {1, ..., d}, if g ∈ B • (f, f,ε ), then for every

x j ∈ [ε, 1 -ε], x i ∈ [0, 1], i ∈ {1, ..., d} \ {j} we have (7) |∂ j,± g(x 1 , ..., x d ) -∂ j f (x 1 , ..., x d )| < ε. Proof. It is enough to fix one j ∈ {1, ..., d}. Recall that f ∈ CC d ∩ C 1 ([0, 1] d ) implies that ∂ j f is non-decreasing in x j and is uniformly continuous on [0, 1] d .
Using this uniform continuity, there exists a partition 0 = x j,0 < x j,1 < ... < x j,K = 1 such that x j,1 < ε and for every x i ∈ [0, 1] with i ∈ {1, ..., d} \ {j}, one has for every l = 1, ..., K, (8)

∂ j f (x 1 , ..., x j-1 , x j,l , x j+1 , ..., x d )-∂ j f (x 1 , ..., x j-1 , x j,l-1 , x j+1 , ..., x d ) < ε 4 .

Set (9)

f,ε,j = ε 4 min l=2,...,K (x j,l -x j,l-1 ).

Consider any function g ∈ B • (f, f,ε,j ), and fix x = (x 1 , ...,

x d ) ∈ [0, 1] d , so that x j ∈ [ε, 1 -ε].
There exists an integer l ∈ {1, ..., K -1} such that x j ∈ [x j,l , x j,l+1 ]. Set x(l ) = (x 1 , ..., x j-1 , x j,l , x j+1 , ..., x d ).

Since the two functions t → g(x 1 , ..., x j-1 , t, x j+1 , ..., x d ) and t → f (x 1 , ..., x j-1 , t, x j+1 , ..., x d ) are real convex, one has

∂ j,-g(x) ≥ ∂ j,-g(x(l)) ≥ g(x(l)) -g(x(l -1)) x j,l -x j,l-1 ≥ f (x(l)) -f (x(l -1)) -2 f,ε,j x j,l -x j,l-1 ≥ f (x(l)) -f (x(l -1)) x j,l -x j,l-1 - ε 2 ≥ ∂ j f (x(l -1)) - ε 2 . (10)
Thanks to the convexity of t → f (x 1 , ..., x j-1 , t, x j+1 , ..., x d ), equation (8) gives

∂ j f (x) ≤ ∂ j f (x(l + 1)) < ∂ j f (x(l -1)) + 2 ε 4 .
Hence, one can continue (10) to obtain

∂ j,-g(x) > ∂ j f (x) - ε 2 - ε 2 = ∂ j f (x) -ε.
Moreover, a similar argument can show that

∂ j,+ g(x) < ∂ j f (x) + ε.
This ends the proof, since ∂ j,-g(x) ≤ ∂ j,+ g(x).

The conclusion follows by taking f,ε = min( f,ε,j : j = 1, ..., d).

The one-dimensional version of Lemma 3 is stated as follows.

Lemma 4. Suppose f ∈ CC 1 ∩ C 1 ([0, 1]). For ε > 0 there exist ε > 0 and ε,f > 0 such that for any g ∈ B • (f, ε,f ) and x ∈ [ε, 1 -ε] we have (11) |g ± (x) -f (x)| < ε.
Next, one compares the pointwise exponents of a differentiable convex function f and its derivative f . It is a general property that h f (x) ≤ h f (x) -1, for every differentiable f . A surprising property is that equality necessarily holds when f is convex and

h f (x) ∈ [1, 2), Lemma 5. If f is convex and differentiable on (a, b) and h f (x) ∈ [1, 2) for some x ∈ (a, b), then h f (x) = h f (x) + 1. Proof. It is enough to prove that h f (x) ≤ h f (x) + 1. Since h f (x) ∈ [1, 2), necessarily h = h f (x) < 1. Hence, there exists a sequence (x n ) n≥1 converging to x such that |f (x n ) -f (x)| > |x n -x| h+ 1
n . Without limiting generality, suppose that x n > x. By the monotonicity of f , one has

f (x n ) > f (x) + (x n -x) h+ 1 n . By convexity of f , f (x n ) ≥ f (x) + f (x)(x n -x).
Setting x n = x + 2(x n -x), and using again the convexity of f at x n , one gets

f (x n ) ≥ f (x n ) + f (x n )(x n -x n ) ≥ f (x) + f (x)(x n -x) + (f (x) + (x n -x) h+ 1 n )(x n -x n ) ≥ f (x) + f (x)(x n -x) + (x n -x) h+ 1 n 1 2 (x n -x) = f (x) + f (x)(x n -x) + 1 2 h+1+ 1 n (x n -x) h+1+ 1 n .
This implies h f (x) ≤ h + 1, hence the result.

One investigates what happens for non-differentiable convex functions.

Lemma 6. If f is convex on (a, b) ⊂ R and h f (x) ∈ [1, 2) for some x ∈ (a, b), then min(h f + (x), h f -(x)) ≤ h f (x) -1. Proof. When h f (x) = 1 the lemma is obvious. Set h = h f (x) > 1, and let ε > 0 so that h -ε > 0. By definition, there exists M ∈ R such that one has |f (x + y) -f (x) -M y| ≤ |y| h-ε
for every small y, and there exists a sequence (y n ) n≥1 converging to zero such that

|f (x + y n ) -f (x) -M y n | ≥ |y n | h+ε .
Hence,

|y n | h+ε-1 ≤ f (x + y n ) -f (x) y n -M ≤ |y n | h-ε-1 .
Since the left and right derivatives f + (x) and f -(x) both exist, they both equal M = f (x). Assume, without loss of generality, that there are infinitely many positive y n 's. For every y n ,

f + (x + y n ) -f + (x) ≥ f (x + y n ) -f (x) y n -f + (x) ≥ |y n | h+ε-1 , thus h f + (x) ≤ h + ε -1, which gives the result.
We also prove the following proposition, which somehow asserts that a convex function cannot have exceptional isolated directional pointwise regularity.

For this, consider the d-dimensional unit sphere

S d = {x ∈ R d : x = 1}. Then, we select a finite set of pairwise distinct points (z 1 , z 2 , ...z N ) ∈ (S d ) N for some integer N ≥ 1 such that the convex hull of {z 1 , z 2 , ...z N } contains the d-dimensional ball B(0, 1/2).
Let us choose ε c > 0 so small that :

• 0 < ε c ≤ 1 1000 min( z i -z j , i = j, i, j ∈ {1, ..., N }). • Setting for every i ∈ {1, ..., N } (12) C i = S d ∩ B(z i , ε c ),
then for any choice of z i ∈ C i , the convex hull of {z 1 , z 2 , ...z N } contains the ball B(0, 1/4).

Proposition 7. If h f (x) = h, then there exists i ∈ {1, ..., N } such that for every z i ∈ C i (see (12)), the restriction of f to the straight line passing through x parallel to the vector z i has a pointwise Hölder exponent equal to h.

Proof. Let n be such that n ≤ h < n + 1. We assume without loss of generality that x = 0, f (0) = 0, D k f (0, ..., 0) = 0 for every k ∈ {1, ..., n}. Let ε > 0. By definition, for every x close to 0, |f (x)| ≤ |x| h-ε , and there exists a sequence (

x n = (x n,1 , ..., x n,d )) n≥1 of elements in R d , converging to 0, such that (13) |x n | h+ε ≤ |f (x n )| ≤ |x n | h-ε .
Consider such an element x n , and the sets (C n,i := 4 x n •C i ) i=1,...,N . Let us prove that there exists i n ∈ {1, ..., N } such that for every n ∈ N and

z in ∈ C n,in , |f (z in )| ≥ (|z in |/4) h+ε .
Assume first that for every i ∈ {1, ..., N }, there exists

z i ∈ C n,i such that |f (z i )| < |f (x n )|. By construction, the ball B(0, x n ) is included in the convex hull of these points (z i ) i=1,...,N . By convexity, this would imply that |f (x n )| ≤ max(|f (z i )| : i = 1, ..., N ), hence a contradiction.
Hence, there exists an i n ∈ {1, ..., N } such that for every

z in ∈ C n,in , |f (z in )| ≥ |f (x n )| ≥ |x n | h+ε = (|z in |/4) h+ε .
Turning to a subsequence, one can assume that i n is constant, equal to i ∈ {1, ..., N }. Now, as a consequence of what precedes, for every vector z i ∈ C i , there exists an infinite number of values (

r n = |x n |) n≥1 such that |f (r n z i )| ≥ (|r n z i |/4) h+ε .
Hence, the restriction of f to the straight line passing through 0 parallel to z i has a pointwise Hölder exponent less than h + ε. Since this holds for every ε > 0, and obviously this exponent is bounded below by h (i.e. the exponent of f ), one concludes that this restriction has exactly exponent h.

First typical properties of continuous convex functions

Based on Lemma 3 it is very easy to see that the typical function in CC d is continuously differentiable on (0, 1) d . This was proved in [START_REF] Gruber | Die meisten konvexen körper sind glatt, aber nicht zu glatt[END_REF][START_REF] Howe | Most convex functions are smooth[END_REF] for instance. We give another proof for completeness.

Proposition 8. There is a dense G δ set G in CC d such that every f ∈ G is continuously differentiable on (0, 1) d .
Proof. By convexity, the partial derivatives ∂ j,± f (x) exist for any f ∈ CC d , x ∈ (0, 1) d and j ∈ {1, ..., d}.

Since CC d is separable, one can choose a sequence of convex functions {f m : m = 1, ...} dense in CC d . In addition, by Remark 2, one can assume that all these functions f m are C ∞ ([0, 1] d ) functions.

By uniform continuity of all the partial derivatives of f m , there is a δ n,m > 0 such that for every j, for every x, x ∈ [0, 1] d , ( 14)

|∂ j f m (x) -∂ j f m (x )| < 1 n , when |x -x | < δ n,m .
Applying Lemma 3, it is possible to choose 0

< n,m < 1 n+m such that if f ∈ B • (f m , n,m ) then for every j ∈ {1, ..., d}, if x = (x 1 , ..., x d ) ∈ [0, 1] d with x j ∈ [ 1 n , 1 -1 n ], then (15) |∂ j,± f (x) -∂ j f m (x)| < 1 n .
Let us introduce the sets

G n = ∞ m=1 B • (f m , n,m ) and G = ∞ n=1 G n . It is clear that G is a dense G δ set in CC d .
We prove that any f ∈ G is continuously differentiable. By definition, there exists an infinite sequence of integers (m n ) n≥1 such that f ∈ B • (f mn , n,mn ).

Fix j ∈ {1, .., d}, and focus on the j-th partial derivatives. Combining inequalities ( 14) and (15

), if x, x ∈ [0, 1] d , with x j , x j ∈ [ 1 n , 1 -1 n ] and ||x -x || < n,mn , then |∂ j,± f (x) -∂ j,± f (x )| < |∂ j f mn (x) -∂ j f mn (x )| + 2 n < 3 n .
The ± is the above inequality means that any choice of left or right derivative can be made. From this, letting n → ∞, it follows easily that

∂ j,+ f (x) = ∂ j,-f (x), hence ∂ j f is continuous on (0, 1) d .
However, the next lemma shows that typical functions f in CC d are not differentiable on [0, 1] d . The problem comes from the boundary of the domain.

Proposition 9. There is a dense G δ set G ∞ in G such that every f ∈ G ∞ satisfies the following: for every j ∈ {1, ..., d}, for every x i ∈ [0, 1] with i ∈ {1, ..., d} \ {j}, one has

(16) ∂ j,+ f (x 1 , ..., x j-1 , 0, x j+1 , ..., x d ) = -∞ and (17) ∂ j,-f (x 1 , ..., x j-1 , 1, x j+1 , ..., x d ) = +∞.
Moreover, (18) h f (x 1 , ..., x j-1 , 0, x j+1 , ..., x d ) = 0 = h f (x 1 , ..., x j-1 , 1, x j+1 , ..., x d ).

Proof. We are going to show that for a fixed j, there is a dense G δ set G 0,j in CC d such that if f ∈ G 0,j then (16) and the first equality in (18) holds.

Without loss of generality, one considers j = 1. As in the proof of Theorem 8, one chooses a sequence of C ∞ functions (f m ) m≥1 . We also select integer constants

M m ≥ 1 such that |∂ 1 f m | ≤ M m on [0, 1] d .
For every integer l ≥ 1, let us introduce the mapping ϕ l defined as

ϕ l (x 1 , ..., x d ) = -l l-1 x 1 if 0 ≤ x 1 ≤ l -l , -l -1 if l -l ≤ x 1 ≤ 1.
Clearly, ϕ l is convex and for any fixed n the functions

f m + ϕ n•m•Mm , m = 1, ... are dense in CC d . Set G 0,1 n = ∞ m=1 B • (f m + ϕ n•m•Mm , (n • m • M m ) -n•m•Mm ), and
G 0,1 = ∞ n=1 G 0,1 n .
We prove that if f ∈ G 0,1 , then (18) and hence (16) hold. By definition, there exists an infinite sequence of integers (m n ) n≥1 such that

f ∈ B • f mn + ϕ n•mn•Mm n , (n • m n • M mn ) -n•mn•Mm n .
For ease of notation, set

l n = n • m n • M mn , that is f ∈ B • (f mn + ϕ ln , l -ln n ). Consider x ∈ [0, 1] d , with x 1 = 0. Then for every n ≥ 5, f (0, x 2 , ..., x d ) -f (l -ln n , x 2 , ..., x d ) ≥ (f mn + ϕ ln )(0, x 2 , ..., x d )) -(f mn + ϕ ln )(l -ln n , x 2 , ..., x d )) -2 • l -ln n ≥ -M mn l -ln n + l -1 n -2 • l -ln n = l -1 n 1 - M mn l n l -ln+2 n - 2 l n l -ln+1 n > 1 2l n ,
where we have used the boundedness of ∂f mn by M mn and the fact that l n > > M mn . Hence, for any α > 0 we have

f (0, x 2 , ..., x d ) -f (l -ln n , x 2 , ..., x d ) l -lnα n > 1 2 l lnα-1 n ,
which tends to infinity when n goes to infinity. Hence (16) holds true, and one also deduces that h f (0, x 2 , ..., x d ) = 0.

Similar arguments yield G δ sets G 0,j for j > 1 and G 1,j for j = 1, ..., d such that the functions f in these sets satisfy the corresponding equalities in (16-18).

Finally, the set

G ∞ = d j=1 G 0,j ∩ G 1,j
satisfies the conditions of Proposition 9.

We conclude this section by proving that typical convex functions have only pointwise exponents less than 2.

Proposition 10. There exists a G δ -set G 2 such that for every f ∈ G 2 , E f (h) = ∅ for every h > 2.

Before proving Proposition 10, we introduce some perturbation functions already used in Section 4 of [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF]. Definition 3. For every l ∈ N, the function γ l : [0, 1] → [0, 1] is defined as follows:

• γ l is continuous,

• For every integer j = 0, ...,

2 l 2 -1, if x 1 ∈ [j2 -l 2 , (j + 1)2 -l 2 - 2 -l 4 ], one sets γ l (x 1 ) = j2 -l 2 -l . So γ l is constant on these intervals, • γ l (1) = 2 -l ,
• For every integer j = 0, ..., 2 l 2 -1, the mapping γ l is affine on the intervals [(j + 1)2 -l 2 -2 -l 4 , (j + 1)2 -l 2 ],

These functions γ l are continuous, ranging from 0 to 2 -l , and are strictly increasing only on the 2 l 2 many very small, uniformly distributed, intervals of length 2 -l 4 .

Proof. We start by selecting a set of C ∞ functions {f m : m = 1, 2, ...} which is dense in CC d . We choose an integer M m,3 ≥ 1 such that the second and third partial derivatives ∂ 3 1 f m with respect to the first variable [START_REF] Buczolich | Typical borel measures on [0, 1] d satisfy a multifractal formalism[END_REF] . Observe that the x 1 -partial derivatives ∂ 1 f m of these functions are monotone in the first variable.

x 1 of f m satisfy |∂ 2 1 f m | + |∂ 3 1 f m | ≤ M m,
Then, one introduces the auxiliary functions, depending only on the first variable: for l ≥ 0, (19) γ l (x 1 , x 2 , ..., x d ) = γ l (x 1 ).

The perturbation functions are defined for l ≥ 0 by

(20) f l (x) = f l (x 1 , x 2 , ..., x d ) = x 1 0 γ l (t, x 2 , ..., x d )dt = x 1 0 γ l (t)dt.
Next we apply Lemma 3 to the functions f m + f m+M m,3 +n with ε = ε m,n := 2 -(m+M m,3 +n) 8 and j = 1. There exists a constant, denoted by

m,n > 0, such that if f ∈ B • (f m + f m+M m,3 +n , m,n ), then for any x 1 ∈ [ε m,n , 1 -ε m,n ] and x j ∈ [0, 1] for j = 2, ..., d, one has (21) |∂ 1,± f (x) -∂ 1 (f m + f m+M m,3 +n )(x)| < ε m,n .
Without limiting generality one can assume that m,n ≤ ε m,n . Set

R n = ∞ m=1 B • (f m + f m+M m,3 +n , m,n ). It is not difficult to see that R n is open and dense in CC d . Suppose that G ∞ is the dense G δ set from Proposition 9.
Since G ∞ is a subset of G from Proposition 8 all f ∈ G ∞ are continuously differentiable on (0, 1) d . Moreover, the Hölder exponent is zero of these functions on the boundary of [0, 1] d . Finally, set

G 2 = G ∞ ∞ n=1 R n .
By construction, there exists a sequence of integers (m n ) n≥1 such that f ∈ B • (f mn + f mn+M mn,3 +n , n,mn ) for every n.

For simplification, we set l n = m n +M mn,3 +n, ρ n := ρ mn,n and ε n := ε mn,n , so that for every n ≥ 1,

ρ n ≤ ε n = 2 -(ln) 8 , f ∈ B • (f mn +f ln , n ), and for any x 1 ∈ [ε n , 1 -ε n ] and x j ∈ [0, 1] for j = 2, ..., d (22) |∂ 1,± f (x) -∂ 1 (f mn + f ln )(x)| < ε n .
Proceeding towards a contradiction, suppose that there exists x = (x 1 , x 2 ..., x d ) ∈ [0, 1] d where h f (x) > 2. Since the Hölder exponent of f is zero on the boundary of [0, 1] d , necessarily x ∈ (0, 1) d .

Since h f (x) > 2, one can find ε > 0, D 2 , C x ∈ R such that for every small h,

(23) |f (x + he 1 ) -f (x) -∂ 1 f (x)h -D 2 h 2 | ≤ C x |h| 2+ε .
Without limiting generality we can suppose that ε < 1/2 holds as well.

Consider the unique integer j n ∈ N such that x 1 ∈ [j n 2 -l 2 n , (j n + 1)2 -l 2 n ). Next we consider two cases depending on whether

x 1 ∈ [j n 2 -l 2 n , (j n + 1)2 -l 2 n -2 -l 4 n ], or x 1 ∈ [(j n + 1)2 -l 2 n -2 -l 4 n , (j n + 1)2 -l 2 n ]. Case 1. Assume that x 1 ∈ [j n 2 -l 2 n , (j n + 1)2 -l 2 n -2 -l 4 n ] for infinitely many integers n ≥ 1.
We set 22), ( 23) and the fact that f ∈ B • (f mn + f ln , n ), we obtain

h n = h n e 1 with |h n | = |h n | = 2 -l 2 n /4, such that the first coordinates of x and x + h n both belong to [j n 2 -l 2 n , (j n + 1)2 -l 2 n -2 -l 4 n ]. Combining (
f mn (x + h n ) + f ln (x + h n ) -f mn (x) + f ln (x) -h n (∂ 1 f mn + ∂ 1 f ln )(x) -D 2 h 2 n ≤ C x |h n | 2+ε + 2 n + ε n |h n | ≤ C x |h n | 2+ε + 2 • 2 -(ln) 8 + 2 -(ln) 8 |h n | ≤ (C x + 1)|h n | 2+ε , (24)
where the last inequality holds since

ρ n ≤ ε n ≤ 2 -(ln) 8 < < |h n | 3 for large n.
By using the Taylor polynomial estimate of the C ∞ function f mn , one deduces that

(25) f mn (x+h n )-f mn (x)-∂ 1 f mn (x)h n - ∂ 2 1 f mn (x) 2! h 2 n ≤ M mn,3 3! |h n | 3 .
Since by its definition ∂ 1 f ln (y) = γ ln (x 1 ) (i.e. it is constant) for any y on the line segment connecting x and x + h n , we also have

(26) f ln (x + h n ) -f ln (x) -∂ 1 f ln (x)h n = 0.
Using (24), ( 25) and (26) we infer

∂ 2 1 f mn (x) 2! h 2 n -D 2 h 2 n (27) ≤ (C x + 1)|h n | 2+ε + M mn,3 3! |h n | 3 < (C x + 2)|h n | 2+ε
where, using the fact that ε < 1/2, the last inequality holds if n is sufficiently large since M n,3 ≤ l n ≤ 2 ln ≤ |h n | -1/2 for n large. Now take h n = 8|h n |e 1 . Then ( 24) and (25) used with h n instead of h n for sufficiently large n yield

f ln (x + h n ) -f ln (x) -∂ 1 f ln (x)|h n | + ∂ 2 1 f mn (x) 2! |h n | 2 -D 2 |h n | 2 ≤ M mn,3 3! |h n | 3 + (C x + 1)|h n | 2+ε = |h n | 2+ε ( M mn,3 3! |h n | 1-ε + C x + 1) ≤ |h n | 2+ε (C x + 2). (28)
Now for large n, it follows from (27) and

|h n | < |h n | that f ln (x + h n ) -f ln (x) -∂ 1 f ln (x)|h n | ≤ |h n | 2+ε (2C x + 4). ( 29
)
Next we obtain a contradiction by using a lower estimate of the lefthand side of (29). By convexity of f ln we have ∂ 1 f ln (y) ≥ ∂ 1 f ln (x) when y is on the line segment connecting x and x + h n . Even more, this interval contains a subinterval of length larger than |h n |/8 = |h n | where

∂ 1 f ln (y) = γ ln (y) = ∂ 1 f ln (x) + 2 -l 2 n -ln = γ ln (x 1 ) + 2 -l 2 n -ln .
Thus,

f ln (x + h n ) -f ln (x) ≥ ∂ 1 f ln (x)|h n | + |h n | 8 • 2 -l 2 n -ln .
By (29), one should have

|h n | 8 • 2 -l 2 n -ln ≤ |h n | 2+ε (2C x + 4). Since 8|h n | = |h n | = 2 • 2 -l 2 n we would obtain that 2 -l 2 n -ln ≤ (2 • 2 -l 2 n ) 1+ε (2C x + 4),
a contradiction when n is large.

Case 2. Suppose that x 1 ∈ [(j n + 1)2 -l 2 n -2 -l 4 n , (j n + 1)2 -l 2 n ] for infinitely many n ≥ 1.
We set

h n = h n e 1 with |h n | = |h n | = 2 -l 4 n /2, such that the first coordinates of x and x + h n both belong to [(j n + 1)2 -l 2 n -2 -l 4 n , (j n + 1)2 -l 2 n
]. Since ρ n and ε n are still much smaller than |h n |, equations (24) and (25) still hold, but now (26) is replaced by

(30) f ln (x + h n ) -f ln (x) -∂ 1 f ln (x)h n = ∂ 2 1 f ln (x) 2! h 2 n = 2 l 4 n -ln-l 2 n h 2 n 2 .
Using the same arguments as before but with (30), we deduce that

∂ 2 1 f mn (x) 2! h 2 n -D 2 h 2 n + 2 l 4 n -ln-l 2 n h 2 n 2 < (C x + 2)|h n | 2+ε .
This last inequality becomes impossible when n becomes large, since

| ∂ 2 1 fm n (x) 2
| ≤ M mn,3 ≤ l n ≤ 2 l 3 n . Hence a contradiction.

The upper estimate

We start with the upper bound for the Hausdorff dimensions of the sets

E ≤ f (h). Proposition 11. If 1 < h ≤ 2 and f ∈ CC d , then dim E ≤ f (h) ≤ d + h -2. Proof. It is sufficient to treat the case h ∈ (1, 2). Assume that dim E ≤ f (h) > d + h -2. For every x ∈ E ≤ f (h)
, by Lemma 7, there exists a cone of direction C ix , where i x ∈ {1, ..., d} such that for every z ∈ C ix , the restriction of f to the straight line passing through x parallel to z has exponent less than h.

Let us call E i the set of elements of E ≤ f (h) satisfying this property with i x = i ∈ {1, ..., N }. Obviously, E ≤ f (h) = N i=1 E i , so there exists at least one i ∈ {1, ..., N } such that dim

E i > d + h -2.
Let us recall the following special case of Marstrand's slicing theorem, Theorem 10.10 in Chapter 10 of [START_REF] Mattila | Geometry of Sets and Measures in Euclidean Spaces[END_REF]. Recall that S d is the unit sphere in R d .

Theorem 12. Let E ⊂ [0, 1] d be a Borel set with Hausdorff dimension α ∈ (d -1, d). Then for almost every z ∈ S d (in the sense of (d -1)dimensional "surface" measure), there exists a set E z of positive (d-1)dimensional Hausdorff measure in the hyperplane orthogonal to z such that for every

x ∈ E z , dim E ∩ (x + Rz) = α -(d -1).
Each C i has non-empty interior in the subspace topology of S d , hence it is of positive d -1-dimensional measure. Applying Theorem 12 to

E i , one can find z ∈ C i and x ∈ [0, 1] d such that if D = (x + Rz), then dim E i ∩ D ≥ d + h -2 -(d -1) = h -1.
Let us call g the restriction of f to D. Then g is still a convex function of one variable.

By definition of E i , every x ∈ D ∩ E i satisfies h g (x) ≤ h.

Next, applying Lemma 6 to g, we deduce that min(h

g + (x), h g -(x)) ≤ h -1, for every x ∈ D ∩ E i . holds for all x ∈ [δ 0 , 1 -δ 0 ].
Let us now introduce

G n = n,k B • (f n,m,k , n,m,k ).
By construction, G n is dense in CC 1 . Proposition 8 yields the existence of a dense G δ set D in CC 1 consisting of functions f continuously differentiable on (0, 1).

We finally set

G = D ∞ n=1 G n .
Clearly, G is a dense G δ set in CC 1 .

Suppose that f ∈ G, and set g = f on (0, 1) and

g 0 = g| [δ 0 ,1-δ 0 ] . Then g 0 ∈ ∞ n=1 G M,δ 0 n and equation (3) implies that for any 1 ≤ h ≤ 2, dim E g 0 (h -1) = h -1. But Lemma 5 yields, E g 0 (h -1) = E f | [δ 0 ,1-δ 0 ] (h), hence dim E f (h) = h -1 for any 1 ≤ h ≤ 2.
Since δ 0 can be chosen arbitrarily small by taking a sequence of δ 0 's tending to zero, we can conclude the proof of the theorem.

Next we turn to the higher dimensional case.

Theorem 15. There is a dense G δ set G in CC d such that for any f ∈ G and

1 ≤ h ≤ 2 , one has dim E f (h) ≥ h + d -2. In addition, E f (h) = ∅ for h > 2, E f (h) ∩ [0, 1] d = ∅ for 0 < h < 1, and ∂([0, 1] d ) = E f (0).
Proof. Now instead of the functions, we can "integrate" the proof used in Section 4 of [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF]. The idea is again to reduce the problem to the one-dimensional case. We select one coordinate direction, for ease of notation the first, the x 1 -axis. We use in our proof the perturbation functions which are constant in the directions of the coordinate axes x j , j = 2, ..., d already used in this paper, given in Definition 3.

Let us select a dense set of C ∞ functions {f m : m = 1, 2, ...} which is dense in CC d . The x 1 -partial derivatives, ∂ 1 f m , of these functions will be denoted by g m . The important feature of these functions g m is the fact that they are monotone increasing in the x 1 -variable. As our example at the beginning of the paper shows (see Remark 1), these functions are not necessarily monotone in the other variables, and this is why one cannot "integrate" simply the MISV genericity results. Now, as in [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF] and in the proof of Proposition 10, we use the functions γ l and the perturbations f l defined in (19) and (20).

Next, apply Lemma 3 to the functions f m + f m+n with ε = ε n+m and j = 1. There exists a constant, denoted by m,n > 0, such that

if f ∈ B • (f m + f m+n , m,n ), then for any x 1 ∈ [ε n+m , 1 -ε n+m ] and x j ∈ [0, 1] for j = 2, ..., d, one has |∂ 1,± f (x) -∂ 1 (f m + f m+n )(x)| < ε n+m , that is, (31) |∂ 1,± f (x) -(g m + γ m+n (x))| < ε n+m holds. Without limiting generality one can assume that m,n → 0 if n is fixed and m → ∞. Set R n = ∞ m=1 B • (f m + f m+n , m,n
G = D ∞ n=1 R n .
Let f ∈ G, and 1 ≤ h ≤ 2. We are going to prove that dim E f (h) ≥ h + d -2, by reducing the argument to a situation already totally taken care of in [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF].

By construction, there exists a sequence of integers (m n ) n≥1 such that f ∈ B • (f mn + f mn+n , n,mn ) for every n.

Set g = ∂ 1 f. By (31), when x 1 ∈ [ε n+mn , 1 -ε n+mn ] and x j ∈ [0, 1] for j = 2, ..., d, one has for

x = (x 1 , x 2 , ..., x d ) (32) |g(x) -(g mn + γ mn+n )(x)| < ε n+mn .
Given 0 < δ 0 < 1/10, choose an integer n 1 such that ε n 1 < δ 0 /100. Select an increasing subsequence (n k ) k≥1 such that the following conditions are fulfilled: put

l k = m n k + n k . One assumes that l k > 2 k , ((l k ) 2 + l k )k + 1 < (l k ) 4 , 2 -((l k-1 ) 2 +l k-1 )(k-1)-1 > 100 • 2 -(l k ) 2 and if D k = 2 (l k ) 2 • 2 -((l k ) 2 +l k )k-2 < 1,
then one also assumes that k is so large that D 1 • • • D k-1 > 2 -l k . These conditions are equations ( 27) and (28) in [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF].

Denote by ϕ k the restriction of g mn k onto [δ 0 , 1 -δ 0 ] × [0, 1] d-1 and by g l k the restriction of γ mn k +n k onto [δ 0 , 1 -δ 0 ] × [0, 1] d-1 . For ease of notation for the restriction of g onto [δ 0 , 1 -δ 0 ] × [0, 1] d-1 we will still use the notation g.

Then g ∈ B • (ϕ k + g l k , ε l k ) for all k = 1, ..., where the ball is taken with respect to the supremum norm in the space of continuous functions monotone in the first variable. Now, we are exactly in the context of our previous article [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF], in which we proved the following sequence of propositions (cf Propositions 13-18 of [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF]). We reproduce the definitions given in [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF] and the associated propositions.

For every h ∈ (1, 2) and k ≥ 2, let

F h-1,k = 2 (l k ) 2 -1 j=0 (j + 1)2 -(l k ) 2 -2 (l k ) 2 +l k h-1 , (j + 1)2 -(l k ) 2 - 1 2 2 (l k ) 2 +l k h-1
.

For h = 1, set F h-1,k = F 0,k as

F 0,k = 2 (l k ) 2 -1 j=0 (j + 1)2 -(l k ) 2 -2 k((l k ) 2 +l k ) , (j + 1)2 -(l k ) 2 - 1 2 2 k((l k ) 2 +l k ) .
One has [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF]:

Proposition 16. For h ∈ [1, 2), let k h = max(3, [1/h] + 2) and

F h-1 = k≥k h F h-1,k ⊂ [δ 0 , 1 -δ 0 ].
For every x ∈ F h-1 × [0, 1] d-1 , h g (x) ≤ h -1.

In particular, dim E g (0) = d -1.

Proposition 17. For h ∈ [1, 2), there exists a probability measure µ h-1 such that µ h-1 (F h-1 × [0, 1] d-1 ) = 1 and for every x ∈ F h-1 × [0, 1] d-1 , (33) lim inf

r→0+ log µ h-1 B(x, r) log r ≥ d + h -2.
Hence, using the mass distribution principle, one deduces from (33) that dim(F h-1 × [0, 1] d-1 ) ≥ d + h -2. From the last two propositions, one deduces that for every h ∈ [START_REF] Bayart | Multifractal spectra of typical and prevalent measures[END_REF][START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF], dim E ≤ g (h -1) ≥ d + h -2. Finally, Proposition 18 of [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF] gives: Proposition 18. For every h > 2, E g (h -1) = ∅.

Finally, we use that the functions f have nowhere a pointwise Hölder exponent greater than 2 (by Proposition 10), and we finish the proof of Theorem 2.

If x ∈ F h-1 × [0, 1] d-1 , then h g (x) ≤ h -1 ∈ [0, 1]. Necessarily, h g (x) + 1 ≤ h f (x) ≤ 2. By Lemma 5, the only possibility is h f (x) = h g (x) + 1 ≤ (h -1) + 1 = h. Hence dim E ≤ f (h) = dim E ≤ g (h -1) ≥ d + h -2. Even more, the same argument as above (Proposition 16 combined with Proposition 11) gives µ h-1 (E ≤ f (h)) > 0. Using the upper bound of Theorem 1, one knows that for every large integer n ≥ 1, dim(E ≤ f (h

-1/n)) ≤ h -1/n + d -2, hence µ h-1 E ≤ f (h -1/n) = 0. Since E f (h) = E ≤ f (h) \ n≥1 E ≤ f (h -1/n),
one deduces that µ h-1 (E f (h)) > 0, which implies that dim E f (h) ≥ d + h -2. Since the converse inequality also holds true, one concludes that dim E f (h) = d + h -2.

2 Figure 1 .

 21 Figure 1. Typical spectra for measures, for MISV and for convex functions

  ). It is not difficult to see that R n is open and dense in CC d . Denote by D a dense G δ set in CC d , which consists of functions differentiable on (0, 1) d , and such that (according to Proposition 10) these functions also have nowhere a pointwise exponent strictly greater than 2.

	Finally, set

Research supported by the Hungarian National Research, Development and Innovation Office-NKFIH, Grant 104178.

Research partly supported by the grant ANR MUTADIS ANR-11-JS01-0009.

Hence, at least one of the two sets E ≤ g + (h -1) and E ≤ g - (h -1) has Hausdorff dimension strictly greater than h -1.

But this is impossible, since both functions g + and g -are monotone, and for such functions, by [START_REF] Buczolich | Multifractal spectrum and generic properties of functions monotone in several variables[END_REF], the Hausdorff dimension of

the pointwise exponent of f at any x ∈ (0, 1) d is necessarily larger or equal than 1. The remaining points are located on the boundary, whose dimension is d -1. And for every h ∈ [0, 1], it is easy to build examples of convex functions such that h f (x) = h for every x satisfying x 1 = 0, so the upper bound d -1 for the Hausdorff dimension of E ≤ f (h) is optimal.

The lower estimate

Using Lemma 4 it is rather easy to "integrate" the result about functions in M 1 = M to obtain the one-dimensional result.

Theorem 14. There is a dense G δ set G in CC 1 such that for any f ∈ G, and for any

Proof. Suppose 0 < δ 0 < 1/2 is fixed.

Recalling the result on typical monotone continuous functions, there exists a G δ set of functions G M,δ 0 in the set M 1,δ 0 := {f :

Let us choose a dense sequence (g n,m ) ∞ m=1 in G M,δ 0 n . By taking antiderivatives of the elements of this sequence and by a suitable definition on the intervals [0, δ 0 ) ∪ (1 -δ 0 , 1], one can choose a sequence of convex functions (f n,m,k ) +∞ m,k=1 which is dense in

where the ball B • is taken in the set M 1,δ 0 using the L ∞ -norm. Lemma 4 gives the existence of n,m,k > 0 such that for every f ∈ B • (f n,m,k , n,m,k ) ⊂ CC 1 , the inequality |f ± (x) -g n,m (x)| < ε n,m


S T Éphane

Xiaochuan Yang
email: xiaochuan.j.yang@gmail.com

Stéphane Seuret
email: seuret@u-pec.fr

ON SOJOURN OF BROWNIAN MOTION INSIDE MOVING BOUNDARIES

We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension and upper mass dimension of these sets are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.

Introduction

This article is concerned with the sojourn properties of the one-dimensional Brownian motion {B t : t ≥ 0} within some moving boundaries. More precisely, for an appropriate function ϕ : R + → R + , consider the sets

E(ϕ) = {t ≥ 0 : |B t | ≤ ϕ(t)} . (1)
that we call the set of Brownian sojourn within two sided boundary ϕ(•).

Besides its obvious application in physics and finance, the understanding of these sets at different scales entails considerable information on the path properties of the Brownian motion. Two types of study are of particular interest.

• Geometric properties of E(ϕ) near t 0 = 0. This corresponds to the regular behavior of a Brownian path near zero. Concretely, local asymptotics of the Brownian motion such as Khintchine's law of iterated logarithm can be described in terms of geometric properties of these sets around 0 with specific choices for ϕ. A natural question is under which condition on ϕ these sets admit an upper density with respect to Lebesgue measure (denoted by | • | throughout the paper), i.e. Uchiyama [START_REF] Uchiyama | The proportion of Brownian sojourn outside a moving boundary[END_REF] treated the case ϕ(t) = h(t) √ t where h is taken from a whole class of correction functions (of logarithmic order), and he found the precise value of the constant c ϕ .

• Geometric properties of E(ϕ) at infinity. This is related to the long time behavior of the Brownian motion. As {B t : t ≥ 0} behaves like a square root function at infinity (in expectation), the set E(ϕ), when ϕ grows slower than the square root function, concerns the lower than normal growth of the Brownian motion. Uchiyama [START_REF] Uchiyama | The proportion of Brownian sojourn outside a moving boundary[END_REF] established upper density bounds for ϕ(t) = √ t/h(t) with h belonging to a large class of correction functions with logarithmic order.

Let us mention briefly some related work. Consider

E(ϕ) = {t ≥ 0 : |B t | ≥ ϕ(t)}.
where ϕ grows like a square root function with a logarithmic order correction. The geometry of these sets around zero describes the local behavior of Brownian motion, whereas their geometry around infinity describes the distribution of high peaks of the Brownian motion -we refer the interested reader to Strassen [START_REF] Strassen | An invariance principle for the law of the iterated logarithm[END_REF] and Uchiyama [START_REF] Uchiyama | The proportion of Brownian sojourn outside a moving boundary[END_REF] for upper density results (around zero and Date: June 13, 2017. infinity). We also mention the recent work of Khoshnevisan, Kim and Xiao [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF] and Khoshnevisan and Xiao [START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF] who consider, among other things, the high peaks of symmetric stable Lévy processes. We refer the reader to [START_REF] Ciesielski | First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path[END_REF][START_REF] Ray | Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion[END_REF][START_REF] Berman | Extreme sojourns of diffusion processes[END_REF][START_REF] Berman | Spectral conditions for sojourn and extreme value limit theorems for Gaussian processes[END_REF][START_REF] Beghin | How the sojourn time distributions of Brownian motion are affected by different forms of conditioning[END_REF] for other results on sojourn properties of stochastic processes.

Motivated by these studies, we focus on the asymptotics around infinity of the sojourn sets of Brownian motion within moving boundaries with much lower than normal growth. For this, we introduce the sets ∀ γ ∈ [0, 1/2], E γ := E(ϕ γ ) with ϕ γ (t) = t γ , and our goal is to estimate the size of E γ for all values of γ. The "size" is expressed in terms of the notion of large scale dimensions developed by Barlow and Taylor [START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF][START_REF] Barlow | Defining fractal subsets of Z d[END_REF] in the late 80's, recently "refreshed" in the work of Xiao and Zheng [START_REF] Xiao | Discrete fractal dimensions of the ranges of random walks in Z d associate with random conductances[END_REF], Georgiou et al. [START_REF] Georgiou | The dimension of the range of a random walk[END_REF], Khoshnevisan et al. [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF][START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF].

The initial motivation of Barlow and Taylor was to define a notion of fractals in discrete spaces such as Z d . This permits to describe the size properties of models in physical statistics, such as the infinite connected component of a percolation process, the range of a transient random walk for instance. To this end, they introduced and investigated several notions of dimension describing different types of asymptotics of a set around infinity. Each dimension corresponds to an analogue in large scales of a classical fractal dimension. Among these dimensions, we are going to use the "macroscopic Hausdorff dimension" and the "mass dimensions", which are respectively the analogs of the classical Hausdorff and Minkowski dimensions in large scales.

Let us state our main result. The macroscopic Hausdorff dimension of a set E ⊂ R d is denoted by Dim H E and the upper mass dimension by Dim UM E. Their definitions are recalled in Section 2.

Theorem 1. Almost surely, for all γ ∈ [0, 1/2],

Dim H E γ = 1 2 if γ ∈ [0, 1/2) 1 if γ = 1/2
(2)

Dim UM E γ = 1 2 + γ. (3)
It is quite surprising that the macroscopic Hausdorff dimension of E γ is constant for all γ ∈ [0, 1/2). This can be interpreted by the fact that, from the macroscopic Hausdorff dimension standpoint, B spends most of the time at its "boundary", i.e. the farthest possible from 0.

Further, one notices in the proof below that a.s. the Brownian zero set

Z = {t ≥ 0 : B t = 0},
which is a priori thinner than E 0 (hence than all the E γ 's), also has macroscopic Hausdorff dimension 1/2. The local structure of Z is well understood since the works by Taylor and Wendel [START_REF] Taylor | The exact Hausdorff measure of the zero set of a stable process[END_REF] and Perkins [START_REF] Perkins | The exact Hausdorff measure of the level sets of Brownian motion[END_REF], who proved Hausdorff measure results for Z using local times. In particular, the classical Hausdorff dimension of Z is 1/2, a result already known in [START_REF] Blumenthal | Some theorems on stable processes[END_REF]. Our result gives the large scale structure of Z and might be compared with an interesting result by Khoshnevisan [START_REF] Khoshnevisan | A discrete fractal in Z 1 +[END_REF] who states that the zero set of a symmetric random walk {ξ n : n ∈ N} in Z 1 with finite variance

{n ∈ N : ξ n = 0}
has macroscopic Hausdorff dimension 1/2. Although they have the same macroscopic Hausdorff dimension, the sets E γ differ by their upper mass dimensions Dim UM . Indeed, (3) indicates a multifractal behavior for Dim UM E γ , and also gives a natural example of sets for which the macroscopic Hausdorff dimension and the upper mass dimension differ.

Our contribution to the formula ( 2) is for all γ ∈ [0, 1/2). The case γ = 1/2 is deduced from Theorem 2 in [START_REF] Uchiyama | The proportion of Brownian sojourn outside a moving boundary[END_REF] where Uchiyama obtained that a.s.

lim sup r→+∞ |E(1/2) ∩ [0, r]| r > 0.
This inequality entails that a.s. Dim H E(1/2) = 1 thanks to the following fact proved in [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF] : for any E ⊂ R,

lim sup r→+∞ |E ∩ [0, r]| r > 0 ⇒ Dim H E = 1.
As B is recurrent, it does return inside the boundary ϕ γ infinitely many times, for every γ ∈ [0, 1/2]. In particular, the sets E γ for γ ∈ [0, 1/2] are unbounded. Our result allows to quantify the recurrence and fluctuation properties of Brownian motion at large scales.

Through an equivalent definition for the macroscopic Hausdorff dimension, we also obtain a uniform dimension result for the level sets

Z x = {t ≥ 0 : B t = x}.
Theorem 2. Almost surely, there exists at most one real number x 0 ∈ R such that for every

x ∈ R \ {x 0 }, Dim H (Z x ) = 1/2.
Theorem 2 should be compared with Perkins' uniform (classical Hausdorff) dimension result for these level sets [START_REF] Perkins | The exact Hausdorff measure of the level sets of Brownian motion[END_REF]. Our method leaves the possibility of an exceptional point, that we were not able to dismiss; but we strongly believe that the result is globally uniform. This paper is organized as follows. In Section 2, we recall the definition of large scale dimensions and establish some preliminary estimates on the probability that the Brownian motion belongs to a domain bounded by a moving boundary during some arbitrary time interval, on the Brownian local time increments around infinity, and on the hitting probability of subordinators. The first part of the main theorem (the macroscopic Hausdoff dimension) is proved in Section 3 and the second part (upper mass dimension) in Section 4. Theorem 2 is proved in Section 5.

Preliminaries

Throughout the paper, c, C are generic positive finite constants whose value may change from line to line. For two families of positive real numbers (a(x)) and (b(x)), the equation a(x) b(x) means that the ratio a(x)/b(x) is uniformly bounded from below and above by some positive finite constant independent of x. Also, P x denotes the law of Brownian motion starting from x ∈ R, and E x denotes the expectation with respect to P x . For simplicity, we also write P = P 0 and E = E 0 . 2.1. Macroscopic dimensions. We adopt the notations in [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF] that we recall now. We will use the notation Q(x, r) = [x, x + r). The length of an interval Q ⊂ R is denoted by s(Q).

Define the annuli ∀ n ≥ 1, S n = [2 n-1 , 2 n ) and S 0 = [0, 1). For any ρ ≥ 0, any set E ⊂ R + , n ∈ N * , we introduce the quantity

ν n ρ (E) = inf m i=1 s(Q i ) 2 n ρ : E ∩ S n ⊂ m i=1 Q i with s(Q i ) ≥ 1 and Q i ⊂ S n . (4)
Other gauge functions could be used instead of x → x ρ , but we will not need them here.

Definition 1. Let E ⊂ R + . The macroscopic Hausdorff dimension of E is defined as

Dim H E = inf    ρ ≥ 0 : n≥0 ν n ρ (E) < +∞    . ( 5
)
The upper and lower mass dimension of E are defined as

Dim UM E = lim sup n→+∞ ln(|E ∩ [0, n]|) ln n , Dim LM E = lim inf n→+∞ ln(|E ∩ [0, n]|) ln n .
The macroscopic Hausdorff dimension of a set does not depend on any of its bounded subsets, since the series in (5) converges if and only if its tail series converges. Further, the covering intervals are chosen to have length larger than 1, which explains why the macroscopic Hausdorff dimension does not rely on the local structure of the underlying set. The same remarks apply to mass dimensions.

It is known [START_REF] Barlow | Fractional dimension of sets in discrete spaces[END_REF][START_REF] Barlow | Defining fractal subsets of Z d[END_REF] that for any set E ⊂ R,

Dim H E ≤ Dim LM E ≤ Dim UM E.
To bound Dim H E from above, one usually exhibits an economic covering of E. To get the lower bound, the following lemma, which is an analog of the mass distribution principle, is useful. Lemma 1. Let E ⊂ S n . Let µ be a finite Borel measure on R with support included in E. Suppose that there exists finite positive constants C and ρ, such that for any interval Q(x, r) ⊂ S n with r ≥ 1, one has µ(Q(x, r)) ≤ Cr ρ .

Then ν n ρ (E) ≥ C -1 2 -nρ µ(S n ). Actually, there is more flexibility in the choice of covering intervals when we are only concerned with the value of the macroscopic dimension. Let us introduce, for every integer n ≥ 1 and any set E ⊂ R + , the quantity

ν n ρ (E) = inf m i=1 s(Q n,i ) 2 n ρ : E ∩ S n ⊂ m i=1 Q n,i with s(Q n,i ) n 2 ∈ N * and Q n,i ⊂ S n . (6)
The difference with ν n ρ is that coverings by sets of size that are multiple of n 2 are used, and this does not change the value of the macroscopic Hausdorff dimension, as stated by the following lemma.

Lemma 2. For every set E ⊂ R + , Dim H E = inf    ρ ≥ 0 : n≥0 ν n ρ (E) < +∞    . (7)
Proof. Let us denote by d the value in the right hand-side of [START_REF] Blumenthal | Some theorems on stable processes[END_REF].

Obviously, for every ρ ≥ 0,

ν n ρ (E) ≥ ν n ρ (E). Hence, recalling (5), Dim H E ≤ d. If Dim H (E) = 1, then (7) holds trivially. Assume thus Dim H (E) < 1. Let ρ > ρ > Dim H (E), and fix n ≥ 1. Choose Q n := {Q i } i=1,...,m as a finite family of intervals such that E ∩ S n ⊂ m i=1 Q i , with s(Q i ) ≥ 1 and m i=1 s(Q i ) 2 n ρ ≤ 2ν n ρ (E).
Define the finite family of intervals

Q n := { Q i } i=1,...,m as follows: Q i is an interval containing Q i , included in S n , whose length is the smallest possible multiple of n 2 . Observe that if s(Q i ) ≥ n 2 , s( Q i ) ≤ 2s(Q i ), while if 1 ≤ s(Q i ) < n 2 , s( Q i ) ≤ n 2 s(Q i ). By construction, E ∩ S n ⊂ m i=1 Q i . In addition, since ρ < ρ < 1, m i=1 s( Q i ) 2 n ρ = m i=1:s(Q i )<n 2 s( Q i ) 2 n ρ + m i=1:s(Q i )≥n 2 s( Q i ) 2 n ρ ≤ m i=1:s(Q i )<n 2 n 2 s(Q i ) 2 n ρ + 2 m i=1:s(Q i )≥n 2 s(Q i ) 2 n ρ ≤ n 2+ρ -ρ 2 n(ρ -ρ) m i=1:s(Q i )<n 2 s(Q i ) 2 n ρ + 2 m i=1:s(Q i )≥n 2 s(Q i ) 2 n ρ .
When n becomes large,

n 2+ρ -ρ 2 n(ρ -ρ) ≤ 1, hence m i=1 s( Q i ) 2 n ρ ≤ 2 m i=1 s(Q i ) 2 n ρ . One deduces that ν n ρ (E) ≤ 2ν n ρ (E). Since n≥1 ν n ρ (E)
< +∞, the series n≥1 ν n ρ (E) also converges, and d ≤ ρ . Letting ρ tend to Dim H (E) yields the result.

Remark 1. The same argument shows that for any C ≥ 1, using coverings of a set E ∩ S n by sets of size larger than C or than Cn 2 instead 1 or n 2 in the definitions (4) or (6), does not change the value of the macroscopic Hausdorff dimension of E.

Hitting probability estimates of Brownian motion inside the moving boundaries.

The following estimate is useful when looking for an appropriate covering of E γ with respect to different large scale dimensions. Lemma 3. Consider an interval Q(a, r) inside S n , i.e. a ∈ S n and a + r ≤ 2 n . For each 0 ≤ γ < 1/2, define the event

A(n, a, r, γ) = {∃ t ∈ Q(a, r) : |B t | ≤ t γ } . (8)
One has

P (A(n, a, r, γ)) ≤ 2 √ π 2 n(γ-1/2) + 4 √ 2π r a 1/2
.

Following a standard vocabulary, the event A(n, a, r, γ) describes the hitting probability of {B t : t ≥ 0} inside the moving boundary. Lemma 3 states that, when r a is small, this hitting probability is controlled by γ, while it behaves like r a 1/2 when r a becomes large. Basic properties of Brownian motion used in the proof of Lemma 3 can be found in [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF] or [START_REF] Mörters | Brownian motion[END_REF].

Proof of Lemma 3 : One has

P(A(n, a, r, γ)) ≤ P inf t∈Q(a,r) |B t | ≤ 2 nγ = P (|B a | ≤ 2 nγ ) + P |B a | > 2 nγ , inf t∈Q(a,r) |B t | ≤ 2 nγ := P 1 + P 2 .
By the self-similarity of B and recalling that a ∈ S n , one obtains

P 1 = P(|B 1 | ≤ a -1/2 2 nγ ) ≤ 2 π a -1/2 2 nγ ≤ 2 √ π 2 n(γ-1/2) .
Using the symmetry of B, one gets

P B a > 2 nγ , inf t∈Q(a,r) B t ≤ 2 nγ = P B a < -2 nγ , sup t∈Q(a,r) B t ≥ -2 nγ .
Thus,

P 2 ≤ 2P B a > 2 nγ , inf t∈Q(a,r) B t ≤ 2 nγ = 2P B a > 2 nγ , inf t∈Q(a,r) (B t -B a ) ≤ 2 2nγ -B a .
Set Bh = B a+h -B a which is a Brownian motion independent of B a . Using successively the self-similarity, the symmetry and the Markov property yields

P 2 ≤ 2P B a > 2 nγ , inf 0≤h≤r Bh ≤ 2 nγ -B a = 2P B a > 2 nγ , inf 0≤h≤1 Bh ≤ r -1/2 (2 nγ -B a ) = 2P B a > 2 nγ , sup 0≤h≤1 Bh ≥ r -1/2 (B a -2 nγ ) = 2 +∞ 2 nγ P sup 0≤h≤1 Bh ≥ r -1/2 (x -2 nγ ) e -x 2 2a dx √ 2πa := I 1 + I 2 ,
where I 1 is the integral between 2 nγ and 2 nγ + r 1/2 , and I 2 is the other part of the integral. On one hand, bounding from above the probability inside the integral by 1, one obtains

I 1 ≤ 2 2 nγ +r 1/2 2 nγ e -x 2 2a dx √ 2πa ≤ 2 √ 2π r a 1/2
.

On the other hand, one knows by applying the reflection principle to B that sup 0≤h≤1 Bh has the same distribution as | B1 |. Hence, using the tail probability estimates of standard Gaussian variable, one has

I 2 ≤ 4 +∞ 2 nγ +r 1/2 P B1 ≥ r -1/2 (x -2 nγ ) e -x 2 2a dx √ 2πa ≤ 4 +∞ 2 nγ +r 1/2 1 r -1/2 (x -2 nγ ) √ 2π e -(x-2 nγ ) 2 2r e -x 2 2a dx √ 2πa ≤ 4 1 √ 2π +∞ 2 nγ e -(x-2 nγ ) 2 2r dx √ 2πa = 4 √ 2π r a 1/2 +∞ 2 nγ e -(x-2 nγ ) 2 2r dx √ 2πr = 2 √ 2π r a 1/2 .
Therefore, one has established that

P 2 ≤ 4 √ 2π r a 1/2 .
Combining the estimates ends the proof.

2.3. Brownian local times and 1/2-stable subordinator. Let us recall very briefly the notion of local times. Let f be a non-negative continuous even function with integral one. For each ε > 0, set

f ε,x (•) = f ((• -x)/ε)/ε. Define L x t = lim ε→0 1 2ε t 0 f ε,x (B s )ds.
P(L t ≥ x) = P(|B 1 | ≥ x/ √ t) ≤ e -x 2 /(2t) . (9)
For each t ≥ 0, define T t = inf{s ≥ t : B s = 0} the first hitting time at zero of Brownian motion after time t. Since L only increases when the Brownian motion hits zero, one has for almost every sample path ω that, L t+h -L t ≤ L Tt+h -L Tt = L h • θ Tt for any t, h ≥ 0, see [17, page 402] for the equality. Here θ is the usual shift operator on the canonical Wiener space. This, combined with (9) and the strong Markov property at the stopping time T t , implies that for every x > 0,

P   sup t≤2 n t∈N sup 2≤h≤2 n-2 h∈N L t+h -L t h/ log 2 h ≥ x   ≤ 2 2n-2 sup t≤2 n 2≤h≤2 n-2 P L t+h -L t h/ log 2 h ≥ x ≤ 2 2n-2 sup 2≤h≤2 n-2 P L h ≥ x h/ log 2 h ≤ 2 2n e -x 2 /(2n)
Taking x = 4n in the above inequality, and summing over n ≥ 1, the Borel-Cantelli lemma yields the conclusion.

It is well known that the right continuous inverse of the Brownian local time at zero {σ t : t ≥ 0} defined by σ t = inf{s ≥ 0 : L s > t} is a 1 2 -stable subordinator. Also, the closure of the range of the associated subordinator coincides with Z, see [START_REF] Bertoin | Subordinators: examples and applications[END_REF] or [START_REF] Marcus | Markov processes, Gaussian processes, and local times[END_REF]. A subordinator {σ t : t ≥ 0} is a Lévy processes with increasing sample paths. It is said to be 1/2-stable if the Laplace transform of σ 1 is e -Φ(λ) with Φ(λ) = (2λ) 1/2 , for all λ > 0.

The renewal function U (x) is the distribution function of the 0-potential measure of {σ t : t ≥ 0}, i.e.

U (x) = E +∞ 0 1 σt≤x dt .
The 0-potential measure of {σ t : t ≥ 0}, denoted by U (dx), characterizes the law of σ in the sense that its Laplace transform is 1/Φ(λ) for all λ > 0. Tauberian theorems provide the relation between the Laplace exponent Φ of a subordinator and its renewal function. More precisely, one has

U (x) 1 Φ(1/x) , (10)
see Section 1.3 in [START_REF] Bertoin | Subordinators: examples and applications[END_REF]. For a general subordinator, one has the following hitting probability estimate in terms of its renewal function. It gives bounds for the probability that its range touches a deterministic set A ⊂ R.

Lemma 5 ([6], Lemma 5.5). For every 0 < a < b, one has

U (b) -U (a) U (b -a) ≤ P R ∩ [a, b] = ∅ ≤ U (2b -a) -U (a) U (b -a) ,
where R = σ [0, +∞) is the range of {σ t : t ≥ 0}.

Next lemma gives a more precise hitting probability estimate for 1/2-stable subordinators.

Lemma 6. Let {σ t : t ≥ 0} be a 1 2 -stable subordinator and Q(a, r) = [a, a + r) be a interval in some annulus S n . One has c r a

1/2 ≤ P R ∩ Q(a, r) = ∅ ≤ C r a 1/2 . ( 11
)
where c, C are independent of a, r and n.

Proof. First, applying [START_REF] Khoshnevisan | A discrete fractal in Z 1 +[END_REF] yields

U ((a + r) -a) = U (r) ≥ Cr 1/2 . ( 12
)
On the other hand, by the definition of U (x), one obtains

U (2(a + r) -a) -U (a) = +∞ 0 P(σ t ∈ [a, a + 2r])dt = +∞ 0 a+2r a p t (x)dx dt
where p t (x) is the density of σ t , given by

p t (x) = C t x 3/2 e -t 2 2x
and C is the normalizing constant such that p t (x) is a probability measure. As Q(a, r) ⊂ S n , r < a and one deduces that

U (a + 2r) -U (a) ≤ C +∞ 0 a+2r a t a 3/2 e -t 2 6a dx dt ≤ C r a 3/2 +∞ 0 te -t 2 /(6a) dt = C r a 1/2 . ( 13
)
The right inequality in [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF] follows then by combining ( 13), [START_REF] Khoshnevisan | On the macroscopic fractal geometry of some random sets[END_REF] and Lemma 5 applied with b = a + r.

Similarly, one gets by ( 10)

U ((a + r) -a) = U (r) ≤ Cr 1/2 ,
and by r < a

U (a + r) -U (a) = +∞ 0 P(σ t ∈ [a, a + r]) dt = +∞ 0 a+r a p t (x)dx dt. ≥ C +∞ 0 a+2r a t a 3/2 e -t 2 2a dx dt = C r a 1/2 . ( 14
)
Applying Lemma 5 with b = a + r and using the last inequalities, one gets the left inequality in [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF].

Proof of Theorem 1 : macroscopic Hausdorff dimension

In this section, we prove the dimension formula [START_REF] Barlow | Defining fractal subsets of Z d[END_REF]. As said in the introduction, by Uchiyama's upper density result [START_REF] Uchiyama | The proportion of Brownian sojourn outside a moving boundary[END_REF], it is enough to compute Dim H E γ for all γ ∈ [0, 1/2).

Let 0 ≤ γ < 1/2 be fixed throughout this section. Due to the monotonicity in γ of the sets E γ , and the fact that the zero set of Brownian motion Z = {t ≥ 0 : B t = 0} is included in E(0), we divide the proof of (2) into two parts :

Dim H E γ ≤ 1 2 and Dim H Z ≥ 1 2 .
Let us start with the upper bound. Let γ ∈ [0, 1/2) and ρ > 1/2 be fixed. Set

x n,i = 2 n-1 + i2 2nγ for i ∈ {0, . . . , 2 n-1 /2 2nγ }.
Consider the intervals Q(x n,i , 2 2nγ ) which form a partition of S n . Note that E γ ∩ Q(x n,i , 2 2nγ ) = ∅ outside of the event A(n, x n,i , 2 2nγ , γ) (see its definition in Lemma 3). Thus,

ν n ρ (E γ ) ≤ 2 (1-2γ)n-1 i=0 2 2nγ 2 n ρ 1 A(n,x n,i ,2 2nγ ,γ) .
By choosing the side length 2 2nγ , one observes that the two terms in Lemma 3 are of the same order. Taking expectation in the above inequality, one obtains by Lemma 3 that there exists a positive finite constant C such that for all n

∈ N * E[ν n ρ (E γ )] ≤ 2 (1-2γ)(1-ρ)n • C2 n(γ-1/2) = C2 (1/2-γ)(1-2ρ)n .
Thus, the Fubini Theorem entails E[ n∈N * ν n ρ (E γ )] < +∞. This proves that Dim H E γ ≤ ρ almost surely. Letting ρ → 1/2 yields the upper bound.

Next we move to the lower bound Dim H Z ≥ 1/2. To this end, we follow the idea of Khoshnevisan [10, page 581].

Let g(ε) = ε log 2 (1/ε) for 0 < ε ≤ 1/2. Let C > 0 be an upper bound for the limsup in Lemma 4. There exists a random integer n 0 (ω) such that for all n ≥ n 0 (ω),

sup t≤2 n t∈N sup 2≤h≤2 n-2 h∈N L t+h -L t h/ log 2 h ≤ 2Cn. ( 15
)
Consider any finite sequence of intervals {Q n,i = [a n,i , b n,i ]} i=1,...,m with integer endpoints and with sidelength 2 ≤ s(Q n,i ) ≤ 2 n-2 , that form a covering of Z ∩ S n .

Using [START_REF] Perkins | The exact Hausdorff measure of the level sets of Brownian motion[END_REF], one obtains

g s(Q n,i ) 2 n = s(Q n,i ) 2 n log 2 2 n s(Q n,i ) = 2 -n/2 s(Q n i ) log 2 s(Q n,i ) (n -log 2 s(Q n i )) log 2 s(Q n,i ) ≥ (n -1) 1/2 2 -n/2 L b n,i -L a n,i 2Cn ≥ n -1/2 2 -n/2 4C (L b n,i -L a n,i ).
where we used the elementary inequality x(n -x) ≥ √ n -1 for all 1 ≤ x ≤ n -1. Therefore, a.s. for all n ≥ n 0 (ω) and any covering of Z ∩ S n by a finite family of intervals

(Q n,i ) i=1,...,m , m i=1 g s(Q n,i ) 2 n ≥ L 2 n -L 2 n-1 4C √ n2 n .
The desired lower bound for Dim H Z follows if we can show that a.s.

F N = N n=1 L 2 n -L 2 n-1 √ n2 n (16)
diverges as N → +∞. Indeed, this would demonstrate that the sum (5) diverges for every ρ < 1/2 and for any choice of covering.

To this end, let us use a second moment argument. As for any fixed t, L t = |B t | in distribution, one knows that

E[L t ] = c √ t and E[(L t ) 2 ] = t. (17)
An Abel summation manipulation gives

F N = L 2 N √ N 2 N - L 2 √ 2 + N n=2 L 2 n 1 √ n2 n - 1 (n + 1)2 n+1 .
Thus,

E[F N ] √ N .
By Cauchy-Schwarz inequality, E[F 2 N ] ≥ cN . To deduce an upper bound for E[F 2 N ], we come back to the expression [START_REF] Ray | Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion[END_REF]. Recall that T t is the first time the Brownian motion hits zero after time t and L t+h -L t ≤ L Tt+h -L Tt = L h • θ Tt . By the strong Markov property applied at the stopping times T 2 n-1 , T 2 -1 , T 2 j-1 (for the integers j and defined below), one has using ( 17)

E[F 2 N ] = N n=1 E[(L 2 n -L 2 n-1 ) 2 ] n2 n + 2 N =1 -1 j=1 E[(L 2 j -L 2 j-1 )(L 2 -L 2 -1 )] j2 j 2 ≤ N n=1 1 n + 2c N =1 -1 j=1 (j ) -1/2 ≤ 3cN.
Therefore, applying Paley-Zygmund inequality yields that there are constants c, c > 0 for which for every N ≥ 1,

P(F N ≥ c √ N ) ≥ c .
This proves P(F ∞ = ∞) > 0. By Kolmogorov's zero-one law, F ∞ = ∞ almost surely, which completes the proof.

Proof of Theorem 1 : upper mass dimension

In this section, we prove that almost surely, for all γ ∈ [0, 1/2],

Dim UM E γ := lim sup

n→+∞ ln |E γ ∩ [0, n]| ln n = 1 2 + γ.
We are going to show that the sojourn time |E γ ∩ S n | is larger than 2 n(1/2+γ) for infinitely many integers n, but at the same time never exceeds 2 n(1/2+γ+ε) for all large n, where ε > 0 can be chosen arbitrarily small. As the underlying sequence of sojourn time are dependent random variables, the following strong law of large numbers for dependent events, proved in Xiao and Zheng [START_REF] Xiao | Discrete fractal dimensions of the ranges of random walks in Z d associate with random conductances[END_REF], is needed.

Lemma 7. [21, Lemma 2.9] Suppose that {A k } k≥1 and {D k } k≥1 are two sequences of events adapted to the same filtration {F k } k≥1 and are such that for some positive constants p, a and δ ∀ k ≥ 1, P(A k+1 |F k ) ≥ p on the event D k , and P(D c k ) ≤ ae -δk . Then there exists ε > 0 such that almost surely,

lim inf n→+∞ n k=1 1 A k n ≥ ε.
Let us first prove the upper bound Dim UM E γ ≤ 1 2 + γ. For each annulus S n , consider the unit intervals Q(k, 1) with left endpoint k ∈ S n .

Recall the definition of the events A(p, k, 1, γ) in Lemma 3. We observe that the union of those intervals Q(k, 1) inside each S p (0

≤ p ≤ n) satisfying 1 A(p,k,1,γ) = 1 forms a covering of E γ ∩ [0, 2 n ]. This implies that |E γ ∩ [0, 2 n ]| ≤ n p=0 2 p -1 k=2 p-1 1 A(p,k,1,γ) .
Taking expectation, one obtains by Lemma 3 that for some finite positive constant C,

E[|E γ ∩ [0, 2 n ]|] ≤ C n p=0 2 p 2 p(γ-1/2) = C2 n(γ+1/2) .
For ρ > 1/2 + γ, the first moment Markov inequality yields that for all n ∈ N * ,

P (|E γ ∩ [0, 2 n ]| ≥ 2 nρ ) ≤ C -1 2 -n(ρ-γ-1/2)
which is the general term of a convergent series. Thus, an application of Borel-Cantelli Lemma gives almost surely, for all n large enough,

|E γ ∩ [0, 2 n ]| < 2 nρ .
Hence, for all m large enough, denote by n the unique integer such that m ∈ S n , one gets

ln |E γ ∩ [0, m]| ln m ≤ ln |E γ ∩ [0, 2 n ]| ln 2 n-1 ≤ n n -1 ρ
Taking lim sup entails Dim UM E γ ≤ ρ, almost surely. The desired upper bound follows by letting ρ tend to γ + 1/2. Now we prove the lower bound Dim UM E γ ≥ 1 2 + γ. Let us fix some notations before moving forward.

Define the super-exponential increasing sequence of integers

n 0 = 1, n k+1 = 2 n k for all k ∈ N * . Set S n = S n ∪ S n+1 ∪ S n+2 .
We introduce two sequences of events for k ≥ 1

A k = |E γ ∩ S n k | ≥ K2 n k (γ+1/2) and D k = |B 2 n k +2 | ≤ n k • 2 (n k +2)/2 ,
recalling that 2 n k +2 is the right endpoint of S n k . Here K ≥ 1 is a universal constant which will be chosen later. One observes that when

A k is realized, |E γ ∩S n | ≥ K2 n(γ+1/2) /3 for some n ∈ {n k , n k +1, n k +2}. So if A k is realized for infinitely many k's, ln |Eγ ∩Sn| ln 2 n
≥ γ + 1/2 for infinitely many integers n, from which the result follows.

To this end, we first chek that the events {A k } k≥1 and {D k } k≥1 verify the conditions of Lemma 7, and then apply this lemma.

Let

F k = σ(B s : s ≤ 2 n k +2
). The condition on {D k } k≥1 is easy to check. Indeed, by the selfsimilarity of Brownian motion, the tail probability estimate of a Gaussian variable, and the rapid increasing rate of {n k }, one gets that

P(D c k ) = P(|B 1 | > n k ) ≤ 2 n k √ 2π e -n 2 k /2 ≤ e -k .
It remains to check the condition on {A k } k≥1 . Applying the Markov property of {B t : t ≥ 0} at time 2 n k +2 yields

P(A k+1 |F k ) = g(B 2 n k +2 ), where g(x) = P x 2 n k+1 +2 -2 n k +2 2 n k+1 -1 -2 n k +2 1 |Bt|≤(t+2 n k +2 ) γ dt ≥ K2 n k+1 (1/2+γ) ≥ P x 2•2 n k+1 2 n k+1 -1 1 |Bt|≤t γ dt ≥ K2 n k+1 (1/2+γ) .
Define the x-level set of the Brownian motion

Z x = {t ≥ 0 : B t = x}
and the stopping time

τ k+1 = inf{t ∈ S n k+1 : B t = x}. Since [τ k+1 , τ k+1 + 2 n k+1 ] ⊂ [2 n k+1 -1 , 2 • 2 n k+1 ] under the event {Z x ∩ S n k+1 = ∅}, one has g(x) ≥ P x Z x ∩ S n k+1 = ∅, τ k+1 +2 n k+1 τ k+1 1 |Bt|≤t γ dt ≥ K2 n k+1 (1/2+γ) .
By the strong Markov property of Brownian motion at τ k+1 ,

g(x) ≥ P x Z x ∩ S n k+1 = ∅ × P x 2 n k+1 0 1 |Bt|≤2 (n k+1 -1)γ dt ≥ K2 n k+1 (1/2+γ) := P 1 k (x)P 2 k (x).
We estimate separately the two terms in the last product. Observe that Z x under P x coincides with the range of a 1/2-stable subordinator. So, the hitting probability estimate given by Lemma 6 yields that

P 1 k (x) ≥ C (18)
uniformly for all k. Finally, one observes that uniformly for all

x ∈ [-n k • 2 (n k +2)/2 , n k • 2 (n k +2)/2 ], P 2 k (x) ≥ P 2 n k+1 0 1 |Bt|≤2 (n k+1 -1)γ -x dt ≥ K2 n k+1 (1/2+γ) ≥ P 2 n k+1 0 1 |Bt|≤2 (n k+1 -1)γ-1 dt ≥ K2 n k+1 (1/2+γ) := p k (19)
The sequence {p k } k≥1 is uniformly controlled from below by the following lemma.

Lemma 8. For k ≥ 1, let M k = 2 n k+1 2 n k+1 -1 1 |Bt|≤2 (n k+1 -1)γ /2 dt.
There exist positive constants

C 1 , C 1 , C 2 such that for all k ≥ 1 C 1 2 n k+1 (γ+1/2) ≤ E[M k ] ≤ C 1 2 n k+1 (γ+1/2)
and

E[M 2 k ] ≤ C 2 2 n k+1 (1+2γ) .
Proof. An application of Fubini theorem and the self-similarity of {B t : t ≥ 0} yields the first moment estimate. For the second moment, one has

E[M 2 k ] = 2 2 n k+1 2 n k+1 -1 2 n k+1 t P(|B t | ≤ 2 (n k+1 -1)γ-1 , |B s | ≤ 2 (n k+1 -1)γ-1 )ds dt ≤ 2 2 n k+1 2 n k+1 -1 2 n k+1 0 P(|B t | ≤ 2 (n k+1 -1)γ-1 )P(|B u | ≤ 2 (n k+1 -1)γ )du dt ≤ 2 2 n k+1 2 n k+1 -1 P(|B t | ≤ 2 (n k+1 -2)γ )dt 1+ n k+1 i=1 2 i 2 i-1 P(|B u | ≤ 2 (n k+1 -1)γ )du .
Applying the first moment estimate yields the results.

Applying Paley-Zygmund inequality to p k with K = C 1 /2, one obtains that for all k,

p k ≥ E[M k ] 2 4E[M 2 k ] = C 1 4C 2 > 0.
Combining this, ( 18) and ( 19), one deduces that for all k, on the event D k ,

P(A k+1 |F k ) = g(B 2 n k +2 )≥ C C 1 4C 2 > 0.
Applying Lemma 7, there exists ε > 0 such that almost surely,

lim inf n→+∞ n k=1 1 A k n ≥ ε.
In particular, A k is realized for infinitely many k's. As explained before, this completes the proof for the lower bound.

Proof of Theorem 2 : dimension of all level sets

In the preceding sections, we have proved that almost surely, Dim H (Z 0 ) = 1/2. We are going to prove that almost surely, for every x ∈ R except maybe one point, Dim H (Z x ) = 1/2.

For this, we start with the following easy lemma. For any function f : R + → R and any interval I ⊂ R + , the oscillation of f on I is written

Osc I (f ) = sup I f -inf I f.
Lemma 9. Let {B t : t ≥ 0} be a Brownian motion. With probability one, there exists an integer N ≥ 1 such that for every n ≥ N , for any integer ∈ [2 n , 2 n+1 -n 3/2 ], there exists at least one interval amongst the consecutive intervals {I k := [ + k, + k + 1]} k=0,...,n 3/2 -1 such that the oscillation of B on I k is larger than log log n.

Proof. Fix n ≥ 1. For any integer interval [k, k + 1] ⊂ S n , let us denote pn = P(Osc [k,k+1] B ≥ log log n) > 0. A standard estimate shows that pn = C n e -(log log n) 2 /2 / log log n, where C n is a constant uniformly bounded by above and below with respect to n.

The Markov property implies that the oscillation of Brownian motion on non-overlapping unit intervals are independent and identically distributed. Therefore, for any sequence of n 3/2 /2 consecutive intervals I 1 , ..., I n 3/2 /2 of length 1, the probability that simultaneously the oscillation of {B t : t ≥ 0} on each of these intervals is less than log log n is p n := (1 -pn ) n 3/2 /2 . One sees that p n ∼ e -Cnn 3/2 e -(log log n) 2 /2 / log log n , hence goes fast to 0 (C n is another constant, still bounded away from 0 and ∞).

There are C2 n / n 3/2 /2 disjoint sequences of consecutive n 3/2 /2 integer intervals included in S n . Hence, by independence, the probability that there is at least one such sequence of n 3/2 /2 consecutive intervals such that the oscillation of {B t : t ≥ 0} on all these intervals is less than log log n is

pn := 1 -(1 -p n ) C2 n / n 3/2 /2 , (20)
which tends exponentially fast to 1. The Borel-Cantelli lemma yields that there exists some (random) integer N such that for every n ≥ N , there is always an interval I of length 1 in the n 3/2 /2 consecutive intervals such that the oscillation of {B t : t ≥ 0} on I is larger than log log n.

As a conclusion, with probability one, every interval of length n 3/2 (which contains necessarily n 3/2 /2 consecutive intervals above) in S n contains a subinterval of length 1 on which the oscillation of {B t : t ≥ 0} is larger than log log n.

We now prove that the macroscopic Hausdorff dimension is 1/2 over a semi-infinite interval. Recall that Z x = {t ≥ 0 : B t = x}. Lemma 10. For every x 0 ∈ R, almost surely, Dim H Z x = 1/2 either for all x ≥ x 0 or for all x ≤ x 0 .

Proof. We prove the Lemma for x 0 = 0. One knows that almost surely, Dim H Z 0 = 1/2. First, Lemma 9 gives an integer N such that the conclusions of this lemma hold for n ≥ N . Let Q n = (Q n,i ) i=1,...,mn be a finite family of intervals forming a covering of Z 0 ∩ S n by intervals of diameter multiple of n 2 .

Each Q n,i can be split into a finite number of contiguous intervals Q of length n 2 . Consider one of these intervals Q, and enlarge it by adding on its right side an interval of size n 2 . Call Q = Q + [0, n 2 ] the obtained interval.

If Q intersects Z 0 , by Lemma 9, when n is large, there exists an interval I ⊂ Q of length 1 on which the oscillation of B is greater than log log n. Hence, (at least) one of the two intervals [0, log log n/2] or [-log log n/2, 0] is necessarily included in B Q (the image of Q by {B t : t ≥ 0}). So Q ∩ Z x = ∅, simultaneously for all x ∈ [0, log log n/2] or for all x ∈ [-log log n/2, 0].

We introduce the sets

Q + n,i = Q∈Q n,i : Q∩Z 0 =∅, [0,log log n/2]⊂B Q Q and Q - n,i = Q∈Q n,i : Q∩Z 0 =∅, [-log log n/2,0]⊂B Q Q. ( 21
)
Observe that the two sets Q + n,i and Q - n,i are union of intervals, and that these unions may intersect each other.

Remark 2. Notice that in this situation, for every x ∈ [0, log log n/2] (resp. x ∈ [-log log n/2, 0]), if R n is a covering of Z x ∩Q n,i by intervals I of size that are multiple of n 2 , then

I∈Rn (I +[-n 2 , n 2 ]) necessarily contains Q + n,i (resp. Q - n,i ). Finally we set Q + n = mn i=1 Q + n,i and Q - n = mn i=1 Q - n,i . (22)
Let ρ < 1/2. By definition, almost surely, n≥0 ν n ρ (Z 0 ) = +∞. In order to simplify the notation, we denote for any family

Q n = {Q n,i } i=1,...,mn of intervals included in S n s ρ (Q n ) := mn i=1 s(Q n,i ) 2 n ρ . (23)
By abuse of notation, we use s(E) to denote the diameter of a set E even when E is not an interval. Assume that for every integer n ≥ 1, the family Q n = (Q n,i ) i=1,...,mn is a covering of S n ∩ Z 0 by intervals of length multiple of n 2 and in addition, that Q n is one of the families that minimize the sum s ρ (Q n ) in [START_REF] Bertoin | Subordinators: examples and applications[END_REF], so that

n≥0 s ρ (Q n ) = +∞. ( 24
)
It is obvious from the construction that Z

0 ∩ Q n,i ⊂ Q - n,i ∪ Q + n,i . In addition, it is not possible to have s(Q - n,i ) ρ + s(Q + n,i ) ρ < s(Q n,i ) ρ , otherwise the family Q n would not be a minimizer. Hence max(s(Q - n,i ) ρ , s(Q + n,i ) ρ ) ≥ s(Q n,i ) ρ /2. For the same reason, it is also not possible to decompose Q - n,i (resp. Q + n,i ) into union of intervals {Q -} (resp. {Q + }) (of length multiple of n 2 ) such that max Q + ∈Q + n,i s(Q + ) ρ , Q -∈Q - n,i s(Q -) ρ < s(Q n,i ) ρ /2.
In other words, one necessarily has

max s ρ (Q + n,i ), s ρ (Q - n,i ) ≥ s(Q n,i ) ρ /2.

One deduces by summation over

i that max s ρ (Q + n ), s ρ (Q - n ) ≥ 1 2 mn i=1 s(Q n,i ) 2 n ρ . (25)
Changing a little bit the point of view, considering any set Q + n of intervals covering the set x∈[0,log log n/2] Z x ∩ S n and any set Q - n of intervals covering x∈[-log log n/2,0] Z x ∩ S n , and using the same idea as the one which led to (25), the inequality (25) holds necessarily.

Finally, summing over n, one has max

  n≥1 s ρ (Q + n ), n≥1 s ρ (Q - n )   ≥ 1 2 n≥1 mn i=1 s ρ (Q n ) = +∞,
this being true for any coverings Q + n and Q - n defined as above. One concludes that at least one of the two sums +∞ n=1 s

ρ (Q + n ) or +∞ n=1 s ρ (Q - n ) is +∞. Assume that this is the first one: n≥1 s ρ (Q - n ) = +∞. Let x < 0 and consider Z x = {t ≥ 0 : B t = x}.
Assume also that N (given by Lemma 9) is so large that (log log N )/2 > x.

Let R n be a covering of Z x ∩ S n by intervals R n,i of length that are multiple of n 2 (the sets Q n and Q n,i are used for the optimal covering of Z).

We use Remark 2. By the construction above, necessarily each union of intervals Q - n,i is contained in R n,i ∈Rn (R n,i ± 2n 2 ) (recall that the notation R n,i ± n 2 means that the interval R n,i enlarged by an interval of size 2n 2 on both of its sides). Indeed, using the notations of ( 21), when Q ⊂ R n,i is an interval of size 1 such that the image of Q = Q ± n 2 by B contains [-log log n/2, 0], necessarily Z x ∩ Q = ∅. We deduce that Q - n,i ⊂ R n,i ∈Rn (R n,i ± n 2 ). Denote by R n = R n,i ∈Rn R n,i ± n 2 . Remark 3. Observe that by changing R n,i into R n,i ± n 2 , s ρ ( R n ) is simply s ρ (R n ) multiplied by a factor ∈ [1, 5 ρ ], which will not change the nature (convergence/divergence) of the series involved in the computation of the macroscopic dimension.

The last lines of computations prove that s ρ ( R n ) ≥ 5 -ρ s ρ (Q - n ), which holds for any covering R n of Z x ∩ S n .

We conclude that for any sequence (R n ) n≥1 , with R n a covering of Z x ∩ S n by intervals with length multiple of n 2 (and larger than 5n 2 ), one necessarily has

n≥1 s ρ ( R n ) ≥ 5 -ρ n≥1 s ρ (Q - n ) ≥ 1 2 • 5 ρ n≥1 s ρ (Q n ) = +∞.
Hence, Dim H (Z x ) ≥ ρ. One also sees that the argument holds simultaneously for all x ≤ 0, almost surely.

Finally, applying the previous result to a sequence (ρ n ) n≥1 which tends to 1/2 leads to Dim H (Z x ) ≥ 1/2, for all x ≤ 0. Now we prove Theorem 2.

Proof. The upper bound follows from the fact that for every x ∈ R, Z x is ultimately included in E(γ) for every γ > 0, and Theorem 1 (formula (2)) yields Dim H (Z x ) ≤ 1/2. We turn to the lower bound. Let (x n ) n≥1 be a dense sequence of real numbers. With probability one, the results of Lemma 10 apply to all the x n 's simultaneously. Define Y = sup{y ∈ R : ∀ x n < y, Dim H (Z x ) ≥ 1/2 for all x ≤ x n } with the convention that the supremum of an empty set is -∞. One sees that, almost surely, there are only three possibilities:

(i) Y = +∞, necessarily, for every x ∈ R, Dim H (Z x ) ≥ 1/2;

(ii) Y = -∞, in this case, for each y ∈ R, there exists x n ≥ y such that Dim H (Z x ) ≥ 1/2 for all x ≤ x n . So, for every x ∈ R, Dim H (Z x ) ≥ 1/2; (iii) -∞ < Y < +∞, consequently, for each x n < Y , Dim H (Z x ) ≥ 1/2 for all x ≤ x n , and for each x n > Y , Dim H (Z x ) ≥ 1/2 for all x ≥ x n . This proves that for every x ∈ R \ {Y }, Dim H (Z x ) ≥ 1/2.

Discussions

In this article, we considered the large scale structure of certain sojourn sets where a Brownian motion visits "exceptionally" small values in a large scale dimension sense. Along the way, we use quite specific properties of the Brownian motion. For instance, the explicit probability density function of Brownian motion and 1 2 -stable subordinator are used in Lemma 3 and 6. It would be very interesting to extend the results here to general Lévy processes, where the densities (when they exist) are not explicitly known.

We believe that Theorem 2 holds for all level sets simultaneously. The problem is a lack of symmetry in the current proof. It would be interesting to find an alternative method to remove the unique point of exception (if any) in our result.

  lim sup s→0 |E(ϕ) ∩ [0, s]| s = c ϕ > 0.

  It is known[START_REF] Marcus | Markov processes, Gaussian processes, and local times[END_REF] Lemma 2.4.1] that the convergence occurs uniformly on (t, y) ∈ [0, T ] × [-M, M ], P y almost surely, for any y ∈ R and T, M > 0. There exists a version of the process {L x t ; (t, x) ∈ R + × R} that is jointly continuous[START_REF] Marcus | Markov processes, Gaussian processes, and local times[END_REF] Theorem 2.4.2]. We work with this version in the following. For each fixed x, the process {L x t : t ≥ 0} is called the Brownian local time at x. Below we write L t = L 0 t for simplicity. Note that the non-decreasing continuous process {L t : t ≥ 0} gives rise to a Radon measure whose support is included in the Brownian zero set Z [13, Remark 3.6.2].We will use this measure in an analog of Lemma 1 to study the large scale dimension of Z. Next lemma gives the asymptotic behavior at infinity of the local time increments, which is the continuous analog of Corollary 2.2 in[START_REF] Khoshnevisan | A discrete fractal in Z 1 +[END_REF].

	Lemma 4. One has P-a.s.
	lim sup n→∞	n -1 sup t≤2 n t∈N	sup 2≤h≤2 n-2 h∈N	L t+h -L t h/ log 2 h	< +∞.

Proof. Set S t = sup 0≤s≤t B s . A famous theorem by Lévy

[17, page 240]

states that the processes {S t ; t ≥ 0} and {L t ; t ≥ 0} have the same law. Also, by the reflection principle, L t = |B t | in distribution for any fixed t. Therefore, classical Gaussian tail probability estimates

[13, page 192]

yield that for all t and x > 0,

Acknowledgments

Part of this paper has been written when both authors visited Michigan State University in 2016. The authors wish to thank Prof. Yimin Xiao for his invitation and his insightful remarks on the subject.