Random measures, intersections, and applications

Transcription

1 Random measures, intersections, and applications Ville Suomala joint work with Pablo Shmerkin University of Oulu, Finland Workshop on fractals, The Hebrew University of Jerusalem June 12th 2014

2 Motivation Probabilistic constructions of sets and measures in Euclidean spaces arise in many problems of geometry and analysis. They give rise to random objects which are often more regular than ones obtained via deterministic constructions.

3 Motivation Probabilistic constructions of sets and measures in Euclidean spaces arise in many problems of geometry and analysis. They give rise to random objects which are often more regular than ones obtained via deterministic constructions. We are mainly interested in intersection properties of random (fractal) sets and measures with deterministic sets.

4 Motivation Probabilistic constructions of sets and measures in Euclidean spaces arise in many problems of geometry and analysis. They give rise to random objects which are often more regular than ones obtained via deterministic constructions. We are mainly interested in intersection properties of random (fractal) sets and measures with deterministic sets. According to classical results, if A is one of the following random subsets of R d 1 a Brownian path, 2 a random similar image of a fixed set A 0, 3 fractal percolation limit set, and, if B is a deteministic set, then A B is typically nonempty (and has dimension dim A + dim B d) if dim A + dim B > d.

5 Goals Our goal is to find random sets/measures such that the above claim holds simultaneously for large parametrised families {B t } of sets B t and apply the result for various problems in analysis and geometry:

6 Goals Our goal is to find random sets/measures such that the above claim holds simultaneously for large parametrised families {B t } of sets B t and apply the result for various problems in analysis and geometry: 1 Dimension of non-tube null sets. 2 Uniform Marstrand-Mattila type projection results, without exceptional directions. 3 Excpected intersection size with all algebraic curves, all self-similar sets, etc. 4 Absolute continuity of convolutions of random measures and deterministic measures. two independent random measures. self-convolutions. Applications to sumsets... 5 Fourier decay for random measures. New examples of Salem sets/measures.

7 A class of martingale measures We call a random sequence (µ n : R d [0, + )) of densities a spatially independent (SI) martingale, if 1 µ 0 is a deterministic bounded density with bounded support. 2 There exists an increasing sequence of σ-algebras B n such that µ n is B n -measurable (i.e. µ n (B) is B n - measurable for all Borel sets B). Moreover, for all x R d, E(µ n+1 (x) B n ) = µ n (x). 3 There is C > 0 such that almost surely µ n+1 (x) Cµ n (x) for all n and all x. 4 There is C < such that for any (C2 n )-separated family Q of sets of diameter 2 n, the restrictions {µ n+1 Q B n } are independent.

8 If (µ n ) is an (SI)-martingale, the sequence µ n almost surely converges to a limit measure µ. We are interested in the geometric properties of both (µ n ) and µ.

9 Examples of SI-martingales: Random cut-outs We say that a sequence (µ n ) of measures on R d is of (F, α, η)-cutout type, where F is a collection of compact sets and α, η > 0, if there are numbers δ, C > 0 and a set Ω F, such that µ n = 2 αn 1[Ω \ Mn j=1 Λ(n) j ], (1) for some random subset {Λ (n) j } Mn j=1 of F, and, moreover, there is a finite random variable N 0 such that M n C 2 ηn for all n N 0, and N 0 satisfies the sub-polyomial tail estimate P(N 0 > N) C exp( N δ ).

10 Examples of SI-martingales: Random cut-outs We say that a sequence (µ n ) of measures on R d is of (F, α, η)-cutout type, where F is a collection of compact sets and α, η > 0, if there are numbers δ, C > 0 and a set Ω F, such that µ n = 2 αn 1[Ω \ Mn j=1 Λ(n) j ], (1) for some random subset {Λ (n) j } Mn j=1 of F, and, moreover, there is a finite random variable N 0 such that M n C 2 ηn for all n N 0, and N 0 satisfies the sub-polyomial tail estimate P(N 0 > N) C exp( N δ ). In many cases, e.g. if (Λ n ) n are chosen according to a Poisson point process with a scale and translation invariant intensity measure on R d, these cut-out measures are (SI)-martingales.

11 Ball type cut-outs

12 Ball type cut-outs

13 Ball type cut-outs

14 Ball type cut-outs

15 Ball type cut-outs

16 Ball type cut-outs

17 Ball type cut-outs

18 Cell-type random measures Let F X and τ, η > 0. We say that a sequence (µ n ) of random measures is of (F, τ, η)-cell type, if there are C, δ > 0 such that where: { 1 F (n) j } Mn j=1 M n µ n = c j 1[F j ], (2) j=1 is a random subset of F, 2 The random variables c j = c (n) j j, n almost surely, satisfy 0 c j C 2 τn for all 3 Almost surely there is N 0 such that M n C 2 ηn for all n N 0. Moreover, P(N 0 > N) C exp( N δ ).

19 Cell-type random measures Let F X and τ, η > 0. We say that a sequence (µ n ) of random measures is of (F, τ, η)-cell type, if there are C, δ > 0 such that where: { 1 F (n) j } Mn j=1 M n µ n = c j 1[F j ], (2) j=1 is a random subset of F, 2 The random variables c j = c (n) j j, n almost surely, satisfy 0 c j C 2 τn for all 3 Almost surely there is N 0 such that M n C 2 ηn for all n N 0. Moreover, P(N 0 > N) C exp( N δ ). (SI)-martingales of this type include many natural subdivision random fractals like fractal percolation.

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25 Intersections with deterministic measures Let {η t }, t Γ, be a family of measures indexed by a metric space (Γ, d) and let {µ n } n be an (SI)-martingale. For all t Γ, and n N, we define a measure µ t n as µ t n(a) = A µ n(x) dη t (x), µ t n = µ t n(r d ), and further µ t = lim n µ t n, if the limit exists.

26 Intersections with deterministic measures Let {η t }, t Γ, be a family of measures indexed by a metric space (Γ, d) and let {µ n } n be an (SI)-martingale. For all t Γ, and n N, we define a measure µ t n as µ t n(a) = A µ n(x) dη t (x), µ t n = µ t n(r d ), and further µ t = lim n µ t n, if the limit exists. (Sometimes, we also consider the measures µ t defined as weak limits of µ t n.)

27 Some examples of {η t } Let Ω R d be fixed compact set (the support of µ 0 ) For some 1 k < d, Γ is the subset of affine k-planes which intersect Ω, with the induced natural metric, and η V is k-dimensional Hausdorff measure restricted to V Ω. Given some k N, Γ is the family of all algebraic curves in R 2 of degree at most k which intersect Ω, d is the natural metric, and η γ is length measure on γ Ω. Let m 2, and let Γ be a totally bounded subset of uniformly contractive self similar IFSs with m maps. Suppose that each IFS (g 1,..., g m ) Γ satisfies the OSC. The measure η (g1,...,g m) is the natural self-similar measure for the corresponding IFS.

28 Theorem Let {µ n } n N be an (SI)-martingale, and let {η t } t Γ be a family of measures indexed by (Γ, d). We assume that there are positive constants α, s, θ, γ 0 such that s > α and so that the following holds: 1 Γ has finite box-dimension. 2 η t (B(x, r)) = O(r s ) for all t Γ, x R d, and 0 < r < 1. 3 Almost surely, µ n (x) 2 αn for all n N and x R d. 4 Almost surely, there is a (random) integer N, such that µ t n µ u n sup t,u Γ,t u;n N 2 θn d(t, u) γ 0 <. (3) Then there is γ > 0 (depending on all parameters) such that almost surely µ t n converges uniformly in t, exponentially fast, and the function t µ t is Hölder continuous with exponent γ.

29 Suppose we want to apply the previous theorem for a (SI)-martingale of (F, α, η)-cutout or (F, τ, η)-cell type. The assumption (1) is a technical condition that can be verified in most cases. Likewise, it is usually a priori clear what is the optimal Frostman exponent in (2). (3) is saying that dim µ + dim η t > d. The last condition (4) looks a bit technical, but it turns out that it is strongly related to the geometry of the shapes Λ F, more precisely on the modulus of continuity of for the sets Λ F. t η t (Λ),

30 Λ V u Λ V t {d(x, Λ) ε} V t V u

31 Hölder continuity of orthogonal projections Theorem Let µ n be a (SI)-martingale which is of (F, α, η)-cutout type, or of (F, τ, η)-cell type and satisfies µ n (x) 2 αn for all n, x. Let k {1,..., d 1} with α < k. Suppose that for some constants 0 < γ 0, C < for all V A d,k, all ε > 0 and any isometry f which is ε close to the identity, we have H k (V Λ \ f (Λ)) C ε γ 0 for all Λ F. Then there is γ > 0 such that (i) The sequence µ V n := V µ n dh k converges uniformly over all V A d,k. Denote the limit by µ V. (ii) µ V µ W K d(v, W) γ. (iii) For each W G d,d k, the projection P W µ is absolutely continuous. Moreover, if we denote its density by f W, then the map (x, W) f W (x) is Hölder continuous with exponent γ.

32 Projection theorem without exceptional directions Corollary Let A = spt µ. Under the assumptions of the previous theorem, and almost surely on µ 0, P V (A) has nonempty interior for all V G d,d k.

33 Projection theorem without exceptional directions Corollary Let A = spt µ. Under the assumptions of the previous theorem, and almost surely on µ 0, P V (A) has nonempty interior for all V G d,d k. The above corollary concerns the case dim H µ > d k. If α k, we show that almost surely, dim(p V µ, x) d α for all V G d,d k and P V µ -almost all x V.

34 Projection theorem without exceptional directions Corollary Let A = spt µ. Under the assumptions of the previous theorem, and almost surely on µ 0, P V (A) has nonempty interior for all V G d,d k. The above corollary concerns the case dim H µ > d k. If α k, we show that almost surely, dim(p V µ, x) d α for all V G d,d k and P V µ -almost all x V. In particular, almost surely on µ 0, we have dim H (P V (A)) d α for all V.

35 Projection theorem without exceptional directions Corollary Let A = spt µ. Under the assumptions of the previous theorem, and almost surely on µ 0, P V (A) has nonempty interior for all V G d,d k. The above corollary concerns the case dim H µ > d k. If α k, we show that almost surely, dim(p V µ, x) d α for all V G d,d k and P V µ -almost all x V. In particular, almost surely on µ 0, we have dim H (P V (A)) d α for all V. Under slightly stronger assumptions we can show that a.s dim B (A V) k α for all V G d,d k. and with positive probability, dim H (A V) = k α for an open set of V:s.

36 For fractal percolation in R 2, the above results are known to hold (Falconer-Grimmett 1992, Rams-Simon 2013, 2014, Peres-Rams). Note however, that for fractal percolation, the horisontal and vertical directions are exceptional for the absolute (and Hölder) continuity of the projections.

37 Intersections with more general families of sets/measures Theorem If d = 2 and (µ n ) is a both an (SI)-martingale, and of cut-out type such that each Λ F is a similar image of the Von Koch snowflake, then all the statements of the previous theorem hold if G d,d k is replaced by any family V of algebraic curves of bounded degree.

38 Intersections with more general families of sets/measures Theorem If d = 2 and (µ n ) is a both an (SI)-martingale, and of cut-out type such that each Λ F is a similar image of the Von Koch snowflake, then all the statements of the previous theorem hold if G d,d k is replaced by any family V of algebraic curves of bounded degree. In particular, if dim µ > 1, then a.s. each polynomial image P(A), P: R 2 R, contains an interval.

39 Intersections with more general families of sets/measures Theorem If d = 2 and (µ n ) is a both an (SI)-martingale, and of cut-out type such that each Λ F is a similar image of the Von Koch snowflake, then all the statements of the previous theorem hold if G d,d k is replaced by any family V of algebraic curves of bounded degree. In particular, if dim µ > 1, then a.s. each polynomial image P(A), P: R 2 R, contains an interval. Theorem The following holds almost surely, if µ n is both an (SI)-martingale and of cut-out type, such that each Λ F is a ball: If the almost sure dimension of µ (and A = spt µ) is s, then a.s. for all self-similar sets E R d satisfying the open set condition, we have dim B (E A) max{0, dim E + s d}.

40 Products, convolutions, etc. Under various (but rather general), geometric conditions of the shapes of Λ F, we show that for our random measures µ (with a.s. dimension s (0, )):

41 Products, convolutions, etc. Under various (but rather general), geometric conditions of the shapes of Λ F, we show that for our random measures µ (with a.s. dimension s (0, )): For any deterministic measure ν on R with ν(b(x, r)) = O(r t ) for some t > 1 s, the convolutions µ (rν) are absolutely continuous for all r 0 and if f r denotes the density of µ rν, the map (x, r) f r (x) is locally Hölder continuous. In particular, if dim H E > 1 s, then a.s. all the sets A + re, r 0 contain an interval.

42 Products, convolutions, etc. Under various (but rather general), geometric conditions of the shapes of Λ F, we show that for our random measures µ (with a.s. dimension s (0, )): For any deterministic measure ν on R with ν(b(x, r)) = O(r t ) for some t > 1 s, the convolutions µ (rν) are absolutely continuous for all r 0 and if f r denotes the density of µ rν, the map (x, r) f r (x) is locally Hölder continuous. In particular, if dim H E > 1 s, then a.s. all the sets A + re, r 0 contain an interval. If (µ ) and (µ ) are independent realisations of the same random measure in R d with s > d/2, the same holds a.s for all µ S(µ ), S GL(R d ).

43 Products, convolutions, etc. Under various (but rather general), geometric conditions of the shapes of Λ F, we show that for our random measures µ (with a.s. dimension s (0, )): For any deterministic measure ν on R with ν(b(x, r)) = O(r t ) for some t > 1 s, the convolutions µ (rν) are absolutely continuous for all r 0 and if f r denotes the density of µ rν, the map (x, r) f r (x) is locally Hölder continuous. In particular, if dim H E > 1 s, then a.s. all the sets A + re, r 0 contain an interval. If (µ ) and (µ ) are independent realisations of the same random measure in R d with s > d/2, the same holds a.s for all µ S(µ ), S GL(R d ). If s > d/2, the same holds for all µ S(µ ) apart from S = ri d, r > 0.

44 Products, convolutions, etc. Under various (but rather general), geometric conditions of the shapes of Λ F, we show that for our random measures µ (with a.s. dimension s (0, )): For any deterministic measure ν on R with ν(b(x, r)) = O(r t ) for some t > 1 s, the convolutions µ (rν) are absolutely continuous for all r 0 and if f r denotes the density of µ rν, the map (x, r) f r (x) is locally Hölder continuous. In particular, if dim H E > 1 s, then a.s. all the sets A + re, r 0 contain an interval. If (µ ) and (µ ) are independent realisations of the same random measure in R d with s > d/2, the same holds a.s for all µ S(µ ), S GL(R d ). If s > d/2, the same holds for all µ S(µ ) apart from S = ri d, r > 0.

45 For dyadic subdivision fractals µ n on [0, 1] (including fractal percolation), we show that they are a.s Salem measures: ( ˆµ(ξ) = O (1 + ξ ) σ/2), for any σ < dim H µ.

46 For dyadic subdivision fractals µ n on [0, 1] (including fractal percolation), we show that they are a.s Salem measures: ( ˆµ(ξ) = O (1 + ξ ) σ/2), for any σ < dim H µ. Combining this with the result that if α < 1/2, then µ µ is a.s. locally Hölder (and thus bounded), it follows that these measures satisfy a Fourier restriction estimate fdµ L p (R) C f L 2 (µ) for all p 4. This partially answers a quesion of I. Laba.

47 Further applications Existence of non tube-null sets of given dimension s d 1 (Shmerkin and S. 2012) Orthogonal projections of random covering sets (Chen, Koivusalo, Li, S. 2013) Convolutions of m 3 random measures µ 1,..., µ m. Dimension threshold for the existence of given patterns (all progressions, all angles etc) inside random fractal sets. The last two applications requrire a more general version of the main continuity result, allowing certain degree of dependenced along lower dimensional subplanes. (work in progress)

48 Thank you!