Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet

We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $B =\{B(t)\}_{t \in R_+^N}$: (1) If $\text{dim}_H F \leq d/2$, then $B(F)$ is almost surely a Salem set. (2) If $N \leq d/2$, then with probability one $\text{dim}_H B(F) = 2 \text{dim} F$ for all Borel sets of $R_+^N$, where "$\text{dim}_H$" could be everywhere replaced by the ``Hausdorff,'' ``packing,'' ``upper Minkowski,'' or ``lower Minkowski dimension.'' (3) Let $M_k$ be the set of $k$-multiple points of $B$. If $N \leq d/2$ and $ Nk > (k-1)d/2$, then $\text{dim}_H M_k = \text{dim}_p M_k = 2 Nk - (k-1)d$, a.s. The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If $N>d/2$ then (2) fails to hold. In that case, we establish uniform-dimensional properties for the $(N,1)$-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the sectorial local nondeterminism of the Brownian sheet (Khoshnevisan and Xiao, 2004).


Introduction
Let B = {B(t)} t∈R N + denote the (N, d)-Brownian sheet. That is, B is the N -parameter Gaussian random field with values in R d ; its mean-function is zero, and its covariance function is given by the following: otherwise. (1.1) We have written B(t) in vector form as (B 1 (t), . . . , B d (t)), as is customary.
The (N, d)-Brownian sheet is one of the two natural multiparameter extensions of the ordinary Brownian motion in R d . The other one is Lévy's N -parameter Brownian motion or, more generally, (N, d)-fractional Brownian motion (fBm) of index H ∈ (0, 1).
It has been long known that fractional Brownian motion is locally nondeterministic (LND, see Pitt, 1978) whereas the Brownian sheet B is not. As a result, two distinct classes of methods have been developed; one to study fractional Brownian motion, and the other, Brownian sheet. Despite this, it has recently been shown that the Brownian sheet satisfies the following "sectorial" local nondeterminism (Khoshnevisan and Xiao, 2004): where t 0 k = 0 for every k = 1, . . . , N . Khoshnevisan and Xiao (2004) have applied the sectorial LND of the Brownian sheet to study the distributional properties of the level set Also, they use sectorial LND of the sheet to study the continuity of the local times of B on a fixed Borel set F ⊂ (0, ∞) N . Khoshnevisan and Xiao have suggested that, for many problems, the previously-different treatments of the Brownian sheet and fractional Brownian motion can be unified, and generalized so that they do not rely on many of the special properties of the sheet or fBm.
The present paper is a continuation of Khoshnevisan and Xiao (2004). Our main purpose is to describe how to apply sectorial LND in order to study the geometry of the surface of the Brownian sheet. In some cases, our arguments have analogues for fBm; in other cases, our derivations can be applied to prove new results about fBm; see the proofs of Theorems 3.3 and 3.6, for instance.
First we consider the Fourier dimension of the image B(F ) for a general (N, d)-Brownian sheet, where F ⊂ (0, ∞) N is a fixed Borel set. It is well known that dim H B(F ) = min (d , 2dim H F ) a.s. (1.5) where dim H denotes Hausdorff dimension. If N > d/2 and we replace dim H by the packing dimension dim P , then (1.5) can fail; see Talagrand and Xiao (1996). In rough terms, this is because when N > d/2, dim P B(F ) is not determined by dim P F . It turns out that, in that case, dim P B(F ) is determined by the packing dimension profile of F defined by Falconer and Howroyd (1997); see Xiao (1997) for details.
Clearly, two distinct cases come up in (1.5): dim H F > d/2 or dim H F ≤ d/2. In the first case, Khoshnevisan and Xiao (2004) have shown that B(F ) a.s. has interior points. This verifies an earlier conjecture of Mountford (1989a). Presently, we treat the second case, and prove that for all non-random Borel sets F ⊂ (0, ∞) N with dim H F ≤ d/2, the image B(F ) is almost surely a Salem set with Fourier dimension 2dim H F . That is, When N = 1, B denotes the ordinary Brownian motion in R d , and the latter result is due to Kahane (1985aKahane ( , 1985b, where he also established a similar result for fractional Brownian motion. However, Kahane's proof does not seem to extend readily to the Brownian sheet case. We will appeal to sectorial LND to accomplish this task. Note that the exceptional null-set in (1.5) depends on F . One might ask whether the so-called uniform Hausdorff dimension result is valid. That is, we wish to know whether there exists a single null set outside which (1.5) holds simultaneously for all Borel sets F ⊂ (0, ∞) N . Of course, this can not be true when N > d/2. For instance, consider F to be the zero set B −1 (0). The following establishes this uniform dimension result in the non-trivial case that N ≤ d/2. Its proof can be found in Section 2 below. Theorem 1.3 Choose and fix positive integers N ≤ d/2. Then, where "dim H " can be everywhere replaced, consistently, by any one of the following: "dim H "; "dim P "; "dim M "; or "dim m ." Starting with the pioneering work of Kaufman (1968) for planar Brownian motion, a number of authors have established uniform dimension results for stochastic processes; see Xiao (2004) for a survey of such results for Markov processes and their applications. When N ≤ d/2, the uniform dimension result for the Brownian sheet was first proved by Mountford (1989b). Mountford's proof is based on special properties of the sheet. Lin (1999) has extended the result of Mountford (1989b) to (N, d, α)-stable sheets [see Ehm (1981) for the definition] by using a "stopping time" argument for the upper bound, and by estimating the moments of sojourn times for the lower bound. In Section 3 we provide a relatively elementary proof of Theorem 1.3 which uses our notion of sectorial LND. Our proof is reminiscent of the earlier arguments of Kaufman (1968) and Monrad and Pitt (1987).
As we have mentioned, Theorem 1.3 does not hold when N > d/2. In Section 3 we derive weaker uniform dimension properties for the (N, 1)-Brownian sheet; see Theorems 3.3 and 3.6. Our results are extensions of the results of Kaufman (1989) for one-dimensional Brownian motion.
In Section 4 we determine the Hausdorff and packing dimensions of the set M k of k-multiple points of the (N, d)-Brownian sheet. Thus, we complete an earlier attempt by Chen (1994).
In the above we have mentioned various concepts of fractal dimensions such as Hausdorff, packing and box-counting dimensions, and packing dimension profiles. Xiao (2004) contains a brief introduction on their definitions and properties. We refer to Falconer (1990) for further information on Hausdorff and box-counting dimensions and to Taylor and Tricot (1985) for information on packing measures and packing dimension.
Throughout this paper, B = {B(t)} t∈R N + denotes an (N, d)-Brownian sheet. Sometimes we refer to B as an "(N, d)-sheet," or alternatively a "sheet." We use | · | to denote the Euclidean norm in R m irrespective of the value of the integer m ≥ 1. We denote the m-dimensional Lebesgue measure by λ m . Unspecified positive and finite constants will be denoted by c which may have different values from line to line. Specific constants in Section i will be denoted by c i,1 , c i,2 and so on. Finally, we denote the closed ball of radius r > 0 about x ∈ R d consistently by (1.7)

Salem sets
In this section we continue the line of research of Kahane (1985aKahane ( , 1985bKahane ( , 1993 and study the asymptotic properties of the Fourier transforms of the image measures under the mapping t → B(t), where B is the (N, d)-Brownian sheet. In particular, we will show that, for every non-random Borel set F ⊂ (0, ∞) N such that dim H F ≤ d/2, B(F ) is almost surely a Salem set.
Let ν be a Borel probability measure on R d . We say that ν is an M β -measure if its Fourier transform ν possesses the following property: Note that if β > d/2, then certainly ν is square-integrable on R d . This, and the Plancherel theorem, together imply that ν is supported by a set of positive d-dimensional Lebesgue measure. From this perspective, our main interest is in studying sets that carry only finite M β -measures with β ≤ d/2.
We define the Fourier dimension of ν as Then it is easy to verify that Define the Fourier dimension of a Borel set E ⊂ R d as where P(E) is the collection of all probability measures on E for any Borel set E. The Fourier dimension bears a relation to the Hausdorff dimension of E. First, recall that for every 0 < α < d the α-dimensional Riesz energy of a Borel probability measure ν on R d is a constant multiple of (Kahane, 1985a; Ch. 10). Therefore, the Frostman theorem implies that for every Borel set Moreover, this inequality is often strict, as observed in Kahane (1985, p  (2.7) We point out that O(s, t) is a finite union of hyper-cubes whose sides are parallel to the axes. Moreover, there are no more than n N of these hyper-cubes in O(s, t).
The following lemma is essential for the proof of Theorem 1.2.
Proof O ur proof follows the proof of Proposition 4.2 of Khoshnevisan and Xiao (2004); see Lemma 1.1 of the present paper.
For all u ≥ 0 and 1 ≤ k ≤ N , define Clearly, the process {X k (u)} u≥0 is centered and Gaussian. In fact, a direct computation of its covariance proves that X k is standard Brownian motion.
For all t ∈ [ε, T ] N , we decompose the rectangle [0, t] into the following disjoint union: and ε < s k ≤ t k }, and ∆(ε, t) can be written as a union of 2 N − N − 1 sub-rectangles of [0, t]. Then we have the following decomposition for Here, B (ε, t) := ∆(ε,t) dB 0 (s). Since all the processes on the right-hand side of (2.11) are defined as increments of B 0 over disjoint sets, they are independent. Therefore (2.12) Because {X k 0 (u)} u≥0 is standard Brownian motion and |t j 0 k 0 − s j k 0 | ≥ r for all j = 1, . . . , n, we can apply Equation (8) of Kahane (1985a, p. 266, Eq. (8)) to conclude that [To obtain this, we set Kahane's parameter as follows: γ = n = 1; his p is our n; his t j is our t j k 0 − ε; s is our s k 0 − ε; and his ε is our r.] Our lemma follows from (2.12) and (2.13).
Now we consider the (N, d)-Brownian sheet B. For any Borel probability measure µ on R N + , we let µ B denote the image-measure of µ under the mapping t → B(t). The Fourier transform of µ B can be written as follows: (2.14) The following theorem describes the asymptotic behavior of µ B (ξ) as |ξ| → ∞.
(2.18) By (2.15), we always have Choose and fix some ξ ∈ R d \{0}, and consider r := |ξ| −2 . It follows from Lemma 2.1, the doubling property of function τ , and (2.15) that (2.20) But 1 + ∞ k=1 exp(c 2,6 2 k )c kn 2,3 ≤ c n 2,7 n ρn with ρ := log c 2,3 . Therefore, (2.19) and (2.20) together imply the following bound for the integral of (2.18): Integrate both sides of (2.21) [µ ⊗n (ds)] to find that This and the Stirling formula together imply the existence of an a > 0 such that The Markov inequality implies then that for all b > 0, which is finite as long as we picked b > d/a (N +ρ)/2 . By the Borel-Cantelli lemma, for every Borel set F ⊆ R N + that satisfies dim H F ≤ d/2. To prove the converse inequality, it suffices to demonstrate that if dim H F ≤ d/2 then dim F B(F ) ≥ 2γ a.s. for all γ ∈ (0, dim H F ). Without loss of generality, we may and will assume F ⊂ (0, ∞) N is compact. See Theorem 4.10 of Falconer (1990) for the reasoning. Hence we may further assume that F ⊂ [ε, T ] N for some positive constants ε < T .
Frostman's lemma implies that there is a probability measure µ on F such that µ(U (x, r)) ≤ c 2,11 r γ for all x ∈ R N + and r > 0; see (1.7) for notation. Let µ B denote the image measure of µ under B, and appeal to Theorem 2.2 to find that dim

Uniform dimension results for the images
In this section we prove Theorem 1.3, and present a weak uniform dimension property of the (N, 1)-sheet.
Our proof of Theorem 1.3 is reminiscent of the method of Kaufman (1968) designed for the planar Brownian motion. See also the techniques of Monrad and Pitt (1987) for N -parameter fBm in R d . The following lemma constitutes the key step in our proof; it will come in handy also in Section 4 below. Throughout, we write Lemma 3.1 Choose and fix N ≤ d/2, ε, δ ∈ (0, 1), and β ∈ (1 − δ, 1). Then with probability 1, for all large enough n, there do not exist more than 2 nδd distinct points of the form t j ∈ F n such that Proof T hroughout this proof define Let A n be the event that there do exist more than 2 nδd distinct points of the form 4 −n k j such that (3.2) holds. Let N n be the number of n-tuples of distinct t 1 , . . . , t n ∈ F n such that (3.2) holds; i.e., N n := · · · · · · t 1 ,...,t n ∈Fn all distinct Thus, we estimate Let us fix n − 1 distinct points t 1 , . . . , t n−1 ∈ F n , and first estimate the following sum: For all fixed t 1 , . . . , t n−1 ∈ F n we can find at most (n−1) N points of the form τ u = (τ u 1 , . . . , τ u N ) ∈ [ε, 1] N such that for every = 1, . . . , N , Let us denote the collection of these τ u 's by Γ n := {τ u } u∈U (n) . Clearly, t 1 , . . . , t n−1 are all in Γ n , and #U (n) ≤ (n − 1) N .
It follows from Lemma 1.1 that for every t n / ∈ Γ n , there exists τ un ∈ Γ n such that But the coordinate processes B 1 , . . . , B d are i.i.d. copies of B 0 . Being Gaussian, the conditional density function of B(t n ) given B(t 1 ), . . . , B(t n−1 ) is therefore bounded above by This has content only when t n / ∈ Γ n . If t n ∈ Γ n , then instead we use the obvious bound, The most conservative combination of (3.10) and (3.11) yields (3.12) Note that The last inequality is due to the fact that if N ≤ d/2 then for all fixed τ u , (3.14) Plug (3.13) into (3.12) to obtain We apply induction, and sum the latter over t n−1 , . . . , t 1 , in this order. Thanks to (3.6), this proves that By (3.5) and (3.16), we can bound P(A n ) as follows: We have used also the elementary inequality, According to the Borel-Cantelli Lemma, P(lim sup n A n ) = 0. This finishes the proof of our lemma.
Recall (3.1). For n = 1, 2, . . . and k := (k 1 , . . . , k N ) ∈ 4 n F n define (3.19) Each I n k is then a hyper-cube of side-length 4 −n , and its sides are parallel to the axes. According to Theorem 2.4 of Orey and Pruitt (1973), the Brownian sheet has the same uniform modulus of continuity as Brownian motion, as long as we stay away from the axes. In particular, for all ε ∈ (0, 1), we have, This and Lemma 3.1 together imply our next lemma.
From this, we can readily deduce the following outside a single null set: For all Borel sets Let η ↑ 1/2 and then ε ↓ 0 to find that (3.23) holds also when "dim H " is replaced by either "dim M " or "dim m ." It also holds for "dim P " by regularization of dim M .
To prove the lower bounds it suffices to verify that outside a single null set, for every Borel set E ⊆ R d , where "dim H " could be any one of "dim H ," "dim m ," "dim M ," or "dim P ." Indeed, we can then select E := B(F ) and derive the lower bounds by noticing that B −1 (B(F )) ⊇ F . Equivalently, we seek to prove that for all ε ∈ (0, 1), the following holds a.s., simultaneously for all Borel sets E ⊂ R d : First we prove this for dim H = dim H . By regularization, it suffices to consider only compact sets E ⊂ R d .
Let α > dim H E be fixed (but possibly random); also choose and fix ε, δ ∈ (0, 1) and β ∈ (1−δ, 1). Then we can find balls U (x 1 , r 1 ), U (x 2 , r 2 ), . . . that cover E, and ∞ =1 r α < ∞. As regards the other three dimensions, we note that by Lemma 3.2, for all n large and all Borel sets E ⊂ R d , Here, as before, δ ∈ (0, 1) and β ∈ (1−δ, 1) are fixed. Take the base-4 logarithm of the preceding display, divide it by n, and then apply a standard monotonicity argument to obtain the following: Also define The following lemma is the key to our proof of Theorem 3.3. Sectorial LND plays an important role in its proof.

Lemma 3.4
For all x, y ∈ [ε, 1] N , R > 1 and integers p = 1, 2, . . . , We will estimate the above integral by integrating in the order dt p , dt p−1 , . . . , dt 1 . First let t 1 , . . . , t p−1 ∈ [0, 1] N be fixed and assume, without loss of generality, that all coordinates of t 1 , . . . , t p−1 are distinct. In analogy with (3.3) define We begin by estimating the conditional probabilities Because B 0 is sectorially LND, we have where c 3,9 > 0 is a constant which depends on ε [we have used the fact that x k + t p k ≥ ε for every 1 ≤ k ≤ N ] and where t 0 k = 0 for every k = 1, . . . , N . Observe that for every 1 ≤ k ≤ N , we have where z i,1 k = t i k , z i,2 k = t i k + y k − x k and z i,3 k = t i k + x k − y k for k = 1, . . . , N . It follows from (3.37) and (3.39) that Therefore, we have We note that the points t 1 , . . . , t p−1 introduce a natural partition of [0, 1] N . More precisely, let π 1 , . . . , π N be N permutations of {1, . . . , p − 1} such that for every k = 1, . . . , N , For convenience, we define also t π k (0) k := 0 and t π k (p) k := 1 for all 1 ≤ k ≤ N .
For every t p ∈ [0, 1] N , there is a unique j ∈ {1, . . . , p − 1} N such that t p ∈ I j . Moreover, there exists a point s j (depending on t p ) such that for every k = 1, . . . , N , the k-th coordinate of s j satisfies [If j 1 = 1, then we should also include t p k in the right hand side of (3.44). Since this does not affect the rest of the proof, we omit it for convenience] and for every k = 1, . . . , N . Hence, for every t p ∈ I j , (3.41) can be rewritten as Note that, as t p varies in I j , there are at most 3 N corresponding points s j . Define I G j := t p ∈ I j : for every s j we have |x k − y k | ≤ |t p k − s j k | for all k = 1, . . . , N as the set of "Good" points, and I B j := I j \ I G j be the collection of "Bad points." For every t p ∈ I G j , (3.46) yields If t p ∈ I B j , then for s j satisfying (3.46) we have |t p k − s j k | < |x k − y k | for some k ∈ {1, . . . , N }. We denote the collection of those indices by U. Then, for every k / ∈ U, |x k − y k | ≤ |t p k − s j k |, and we have It follows from (3.47) and (3.48) that I j P(t p ) dt p is at most where the last inequality follows from the facts that |x − y| ≤ √ N and the integral over I B j is bounded by c R −1 . Hence, we have Continue integrating dt p−1 , . . . , dt 1 in (3.34) in the same way, we finally obtain (3.33) as desired.
Remark 3.5 For later use in the proof of Theorem 3.6, we remark that the method of the proof of Lemma 3.4 can be used also to prove that (3.51) In fact, by taking R := 2 7n/8 in (3.41), we obtain We first demonstrate that there exists a constant c 3,15 and an a.s.-finite random variable n 0 = n 0 (ω) such that almost surely for all n > n 0 (ω), (3.55) By choosing p := n, u := c 3,7 8 2N , and owing to Stirling's formula, we know that the probabilities in (3.55) are summable. Therefore, by the Borel-Cantelli lemma, a.s. for all n large enough, Now we are ready to prove (3.53). This is a trivial task unless n 2N 4 −n < |y − x|, which we assume is the case. For x, y ∈ [ε , 1] N , we can findx andȳ ∈ Q n−1 ∩ [ε, 1] N so that |x −x| ≤ √ N 8 −n and |y −ȳ| ≤ √ N 8 −n , respectively. By the modulus of continuity of B 0 , we see that I 2 n (x, y) ≤ I 2 n−1 (x,ȳ) for all n large enough. On the other hand, by (3.56) and the assumption n 2N 4 −n < |y − x|, we have By Theorem 4.10 in Falconer (1990), we may and will assume µ is supported on a compact subset of F . Hence (3.53) is applicable.
Let ν t be the image measure of µ under the mapping x → B 0 (x+t) (x, t ∈ (0, 1] N ). By Frostman's Theorem, in order to prove dim H B 0 (F + t) ≥ η, it suffices to prove that Now we follow Kaufman (1989), and note that the left-hand side is equal to To prove that the last integral is finite for almost all t ∈ [0, 1] N , we integrate it over [0, 1] N and prove that (3.61) We split the above integral over D = {(x, y) : |x − y| ≤ R −2 } and its complement, and denote them by J 1 and J 2 , respectively. Since (µ × µ)(D) ≤ c 3,17 R −2γ and η ∈ (0, 2γ), we have On the other hand, |x − y| −1/2 < R for all (x, y) ∈ D c . Moreover, by (3.53), I R (x, y) < c(ω) (log R) N |x − y| −1/2 . It follows that Again, we will assume that µ is supported by a compact subset of F .
. The sectorial LND of B 0 and (3.72) imply that for all t ∈ S n and ξ ∈ R 2p with |ξ j | ∈ [ 1 2 , 5 2 ] for every 1 ≤ j ≤ 2p, we have  In the above c 3,22 > 0 is a constant depending on ε and again we have used the fact that x + t k ≥ ε for every 1 ≤ ≤ N .
By combining (3.73) and (3.74), we obtain (3.75) Later we will choose the constant c 3,21 such that c 3,23 is sufficiently large. Thus the integral in (3.71) over S n can be neglected. Now, we consider the integral in (3.71) over T n := [0, 1] 2N p \S n , which can be written as (3.76) From (3.76), we can see that T n is a union of at most (4p) 2N p sets of the form: where j := (j ,k : 1 ≤ k ≤ 2p, 1 ≤ ≤ N ) has the property that j ,k = k and where z = 0 or x − y .
The following lemma estimates the Lebesgue measure of T n . Here and throughout, λ denotes the Lebesgue measure on R for all integers ≥ 1.
We now continue with the proof of Lemma 3.7 and defer the proof of Lemma 3.8 to the end of this section.
It follows from (3.76), (3.77) and Lemma 3.8 that We proceed to estimate the integral in (3.71) over T n . It is bounded above by where D n := max Since ψ is a rapidly decreasing function, we derive from (3.79) that  form the maximum chain C 1 . Let Γ 1 be the collection of all s i 's contained in some pairs of C 1 . Secondly we pick a pair (s j , s j ) from the remaining pairs, and let Γ 2 be the collection of all s i 's in the maximal chain containing (s j , s j ). Clearly, Γ 1 ∩ Γ 2 = ∅. Continue this procedure, we get a partition Γ 1 , . . . , Γ L of {s 1 , . . . , s m }, where L is determined by the sequence { 1 , . . . , m }. Denote τ i = #Γ i , so that τ i ≥ 2, L i=1 τ i = m, and L ≤ m/2. With the above partition, we can write wheres i ∈ [0, 1] τ i denotes the vector formed by the elements in Γ i .
In order to estimate the integral on [0, 1] τ i (i = 1, . . . , L), we note that there is a natural tree structure in each Γ i . We pick any element s i 0 ∈ Γ i as the root of the tree; its first generation offspring are the elements s i 01 , . . . s i 0k 1 ∈ Γ i such that for every j = 1, . . . , k 1 either (s i 0 , s i 0j ) or (s i 0j , s i 0 ) is a pair. Similarly, we can find the offspring of each s i 0j , which are the second generation offspring of s i 0 . Continuing this procedure, we obtain a tree associated to Γ i . If this tree has q generations, then we can rewrite the multiple integral over [0, 1] τ i as a τ i -layer iterative integral by integrating the qth generation offspring at first, then the (q − 1)th generation and so on. With the exception of the outside-most layer, each layer of integration contributes a factor 2r. Hence, we have  where the integral on I B j can not be neglected anymore if d > 1. In this general case, we do not know whether Theorems 3.3 and 3.6 remain valid.
The following question was raised by Kaufman (1989) for Brownian motion in R. It is still open, and we reformulate it for the Brownian sheet. Question 3.10 Suppose N > d/2. Is it true that, with probability 1, B(F + t) has interior points for some t ∈ [0, 1] N for every Borel set F ⊂ (0, ∞) N with dim H F > d/2? Xiao (1997) has proved that for the (N, d)-Brownian sheet B with N > d/2 and for any Borel set F ⊆ (0, ∞) N , dim P B(F ) = 2 Dim d/2 F a.s., (3.89) where Dim d/2 F denotes the (d/2)-dimensional "packing-dimension profile" of F defined by Falconer and Howroyd (1997).
In light of (3.89) and Theorem 3.3, we may ask the following natural question:

Hausdorff and packing dimensions of the multiple points
Let k ≥ 2 be an integer. We say that x ∈ R d is a k-multiple point of B if there exist k distinct points t 1 , . . . , t k ∈ (0, ∞) N such that x = B(t 1 ) = · · · = B(t k ). Let M k be the set of k-multiple points of B. Define L k to be the random set, (t 1 , . . . , t k ) ∈ (0, ∞) N k : t i 's are distinct and B(t 1 ) = · · · = B(t k ) . It is possible to show that this formula holds also for dim P L k . Rosen's proof of (4.2) proceeds by studying the regularity of the sample functions of the intersection local times of the sheet.
In the following, we determine the Hausdorff and packing dimensions of M k . An earlier attempt has been made in Chen (1994). But there are gaps in the proof of his Lemma 2.2 [see lines 2-4 and line -6 on page 57].
In Section 3 we have fixed the gaps in Chen's proof. In particular, see Lemma 3.1 above. This is a correct version of Chen's Lemma 2.2. It is proved by appealing to the sectorial LND of the sheet. Thus, we can prove the following. Proof L et L k denote the projection of L k onto R N + ; in particular, note that M k = B( L k ). It follows from Theorem 1.3 and (4.2) that dim P M k ≤ 2N k − (k − 1)d a.s. (4.4) Since dim H M k ≤ dim P M k , it remains to show that dim H M k ≥ 2N k − (k − 1)d a.s. (4.5) For this purpose, let I 1 , . . . , I k be k pairwise disjoint subcubes of [ε, 1] N and let I = k j=1 I j . Define M k (I) to be the collection of all x ∈ R d for which there exists distinct times t 1 , . . . , t k ∈ I such that x = B(t 1 ) = · · · = B(t k ). Thus, M k (I) are the k-multiple points of B which correspond to time-points that are in I. The argument in Chen (1994), coupled with our Lemma 3.1 in place of Chen's Lemma 2.2, correctly implies that P{dim H M k (I) ≥ 2N k − (k − 1)d} > 0. On the other hand, the independent increment property of the Brownian sheet implies a zero-one law for dim H M k ; see pages 110 and 115 of Rosen (1984). Whence, (4.5) follows. This question is closely related to the regularity, in time, of the intersection local times of the sheet. Such regularity theorems are likely to be interesting in their own right. For such results on Brownian motion, see LeGall (1987LeGall ( , 1989.