Spatially independent martingales, intersections, and applications
We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures , and show that under some natural checkable conditions, a.s. the mass of the intersections is Hölder continuous as a function of . This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Łaba in connection to the restriction problem for fractal measures.
Key words and phrases:martingales, random measures, random sets, Hausdorff dimension, fractal percolation, random cutouts, convolutions, projections, intersections
2010 Mathematics Subject Classification:Primary: 28A75, 60D05; Secondary: 28A78, 28A80, 42A38, 42A61, 60G46, 60G57
- 1 Introduction
- 2 Notation
- 3 The setting
- 4 Hölder continuity of intersections
- 5 Classes of spatially independent martingales
- 6 A geometric criterion for Hölder continuity
- 7 Affine intersections and projections
- 8 Fractal boundaries and intersections with algebraic curves
- 9 Intersections with self-similar sets and measures
- 10 Dimension of projections: applications of Theorem 4.4
- 11 Upper bounds on dimensions of intersections
- 12 Lower bounds for the dimension of intersections, and dimension conservation
- 13 Products and convolutions of spatially independent martingales
- 14 Applications to Fourier decay and restriction
1.1. Motivation and overview
One of the most basic examples in the probabilistic method in combinatorics, going back to Erdős’ classical lower bound on the Ramsey numbers , is the random subset of obtained by picking each element independently with the same probability . On one hand, this random set (or suitable modifications) serves as an example in many problems for which deterministic constructions are not yet known, or hard to construct, see . On the other hand, the random sets are often studied for their intrinsic interest. For example, recently there has been much interest in extending results in additive combinatorics such as Szemerédi’s Theorem or Sárkőzy’s Theorem from to the random sets (needless to say, the appropriate choice of function depends on the problem at hand). See e.g. [20, 5] and references therein.
Probabilistic constructions of sets and measures in Euclidean spaces also arise in many problems in analysis and geometry. Again, such constructions are often employed to provide examples of phenomena that are hard to achieve deterministically, and are also often studied for their intrinsic interest. Although there is no canonical construction as in the discrete setting, for many problems (though by no means all) of both kinds one seeks constructions which, to some extent, share the following two key properties of the discrete canonical random set : (i) all elements have the same probability of being chosen, and (ii) for disjoint sets , the random sets are independent. See e.g. [69, 48, 50, 51, 76, 18] for some recent examples of ad-hoc constructions of this kind, meant to solve specific problems in analysis and geometric measure theory. Some classes of random sets and measures that have been thoroughly studied for their own intrinsic interest, and which also enjoy some form of properties (i), (ii) above are fractal percolation, random cascade measures and Poissonian cut-out sets (these will all be defined later). See e.g. [44, 22, 14, 2, 65, 71, 67].
While of course many details of these papers differ, there are a number of ideas that arise repeatedly in many of them. The main goal of this article is to introduce and systematically study a class of random measures on Euclidean spaces which, in our opinion, provides a useful analogue of the canonical discrete random set , and captures the fundamental properties that are common to many of the previously cited works. In particular, this class includes the natural measures on fractal percolation, random cascades, Poissonian random cut-outs, and Poissonian products of cylindrical pulses as concrete examples (see Section 1.2 for the description of two key examples, and Section 5 for the general models). Our main focus is on intersections properties of these random measures (and the random sets obtained as their supports), both for their intrinsic interest, and because they are at the heart of other geometric problems concerning projections, convolutions, and arithmetic and geometric patterns. As a concrete application, we are able to answer a question of Łaba from  related to the restriction problem for fractal measures. We hope that this general approach will find other applications to problems in random structures and geometric measure theory in the future.
To further motivate this work, we recall some classical results. An affine -plane intersects a typical affine -plane if and only if , in which case the intersection has dimension . Here “typical” means an open dense set of full measure (in fact, Zariski dense in the appropriate variety). Similar results, going back to Kakutani’s work on polar sets for Brownian motion, hold when is replaced by a random set sampled from a “sufficiently rich” distribution, by a given deterministic set, and linear dimension by Hausdorff dimension ; here “typical” means either almost surely or with positive probability. For example, if is one of the following random subsets of :
a random similar image of a fixed set (chosen according to Haar measure on the group of linear similitudes),
a Brownian path,
and is a deterministic set, then and intersect with positive probability if and only if ; in the latter case, has dimension at most almost surely, and dimension at least with positive probability (assuming has positive Hausdorff measure in its dimension in (1)). See e.g. [58, 46, 66] for the proofs.
Clearly, one cannot hope to invert the order of quantifiers in any result of this kind, since a set never intersects its complement. Nevertheless, it seems natural to ask whether these results can be strengthened by replacing the fixed set by some parametrized family , and asking if the random set intersects in the expected dimension simultaneously for all (or at least for in some open set) with positive probability. For example, this could be a family of -planes, of spheres, of self-similar sets, and so on. For random similar images of an arbitrary set, it is easy to construct counterexamples, for example for the family of -planes; this is due to the limited (finite dimensional) amount of randomness. Regarding Brownian paths, it is known that if almost surely there are cut-planes, that is, there are planes and times such that and lie on different sides of (where is Brownian motion); in particular, the intersection of these planes with the Brownian path is a singleton. See [13, Theorem 0.6]. On the other hand, some results of this kind for fractal percolation, regarding intersections with lines, have been obtained implicitly in [25, 71, 70]; some of these results will be recalled later. Compared even to Brownian paths, fractal percolation has a stronger degree of spatial independence (as it is not constrained to be a curve), and as we will see this is key in obtaining uniform intersection results with even more general families.
Yet another motivation comes from geometric measure theory. Classical results going back to Marstrand  in the plane and Mattila  in higher dimensions say that for a fixed set , “typical” linear projections and intersections with affine planes behave in the “expected” way. For example, if , then for almost all -planes , the orthogonal projection of onto has positive Lebesgue measure. See [59, 60] for a good exposition of the general theory. Recently there has been substantial interest in improving these geometric results for specific classes of sets and measures, both deterministic (see e.g. [41, 40] and references therein) and, more relevant to us, random (see e.g. [22, 71, 29, 70]).
In this article we introduce a large, and in our view natural, class of random sequences with a limit , which include as special cases the natural measure on fractal percolation, as well as other random cascade measures, and the natural measure on a large class of Poissonian random cutout fractals (see Section 5). The key properties of these measures are inductive versions of the properties (i), (ii) of the canonical random discrete set. We pair these random measures with parametrized families of (deterministic) measures , where is a totally bounded metric space with controlled growth. Examples of families that we investigate include, among others, Hausdorff measures on -planes or algebraic curves, and self-similar measures on a wide class of self-similar sets.
Our main abstract result, Theorem 4.1, says that under some fairly natural conditions on both the random measures and the deterministic family, the “intersection measures” are well defined, and behave in a Hölder-continuous way as a function of ; in particular, with positive probability they are non-trivial for a nonempty open set of . (Some extensions of this theorem are presented in Section 13.) The hypotheses of Theorem 4.1 can be checked for many natural examples of random measures and parametrized families of measures ; more concrete sufficient geometric conditions are provided in Section 6. From Section 7 on, we start applying Theorem 4.1 in different settings, and deducing a large variety of applications.
Before we proceed with the details, we summarize some of our results, and how they relate to our motivation (as described above), and to existing work in the literature.
The projections of planar fractal percolation to linear subspaces (as well as some classes of non-linear projections) were investigated in [71, 70, 77]. Among other things, in those papers it is proved that, when the dimension of the percolation set is , then all orthogonal projections onto lines have nonempty interior, and when the dimension is , then all projections have the same dimension as . We prove that this behavior holds for a large class of natural examples, in arbitrary dimension, including much more general subdivision random fractals and many random cut-outs (see Theorems 7.1 and 10.1). As indicated in our motivation, these are considerable strengthenings, for this class of sets, of the Marstrand-Mattila Projection Theorem.
Moreover, in the regime when the dimension is , we show that for the same class of measures, almost surely the intersection with all lines has box dimension at most (with uniform estimates); see Section 11. Moreover, we show that with positive probability (and full probability on survival of a natural measure), in each direction there is an open set of lines which intersect it in Hausdorff dimension at least (and therefore exactly) ; see Section 12. Note that this is much stronger than asserting that the projections have nonempty interior. Moreover, our results hold for any ambient dimension and intersections with -planes, for any . Furthermore, we prove similar results for intersections with large classes of self-similar sets, and (in the plane) also with algebraic curves. In particular, these results apply to fractal percolation and random cut-outs, thereby extending Hawkes’ classical intersection result  from a single set to natural families of sets as well as the results of Zähle , who considered intersections of random cut-outs with a fixed -plane.
Peres and Rams  proved that for the natural measure on a fractal percolation of dimension in the plane, all image measures under an orthogonal projection are absolutely continuous and, other than the principal directions, have a Hölder continuous density. For the principal directions, the density is clearly discontinuous, and a similar phenomenon occurs for more general models defined in terms of subdivision inside a polyhedral grid. This led us to investigate the following question: given , does there exist a measure supported on a set of Hausdorff dimension , such that all orthogonal projections of have a Hölder continuous density? We give a strong affirmative answer; there is a rich family of such measures, including many arising from Poissonian cut-out process and percolation on a self-similar tiling. The cut-out construction works in any dimension and we obtain joint Hölder continuity in the orthogonal map as well, see Theorem 7.1. Furthermore, we look into the larger class of polynomial projections, and establish the existence of a measure on supported on a set of any dimension , , with the property that all polynomial images are absolutely continuous with a piecewise locally Hölder density. See Theorem 8.5.
Closely related to the size of slices is the concept of dimension conservation, introduced by Furstenberg in ; roughly speaking, a Lipschitz map is dimension conserving if any loss of dimension in the image is compensated by an increase in the dimension of the fibres. We prove that affine and even polynomial maps restricted to many of our random sets are dimension conserving in a very strong fashion. In particular, we partially answer a question of Falconer and Jin . See Section 12.2.
Another problem that has attracted much interest concerns understanding the geometry of arithmetic sums and differences of fractal sets. This is motivated in part by Palis’ celebrated conjecture that typically, if are Cantor sets in the real line with , then their difference set contains an interval. Although the conjecture was settled by Moreira and Yoccoz  in the dynamical context most relevant to Palis’ motivation, much attention has been devoted to its validity (or lack thereof) for various classes of random fractals, see e.g. [22, 23, 21] and the references there. We prove a very strong version of Palis’ conjecture when are independent realizations of a large class of random fractals in , including again many Poissonian cut-outs, as well as subdivision-type random fractals which include fractal percolation as a particular case. Namely, we show that under a suitable non-degeneracy assumption, if , then has nonempty interior for all simultaneously. In fact, we deduce this from an even stronger result about measures: if are independent realizations of a random measure (which again may come from a Poissonian cut-out or repeated subdivision type of process), and the supports have dimension , then the convolution is absolutely continuous with a Hölder density (and also jointly Hölder in ). These results are presented in Section 13.3. See also Theorem 13.1 for a result on the arithmetic sum of a random set and an arbitrary deterministic Borel set.
It has been known since Wiener and Wintner  that there are singular measures such that the self-convolution is absolutely continuous with a bounded density. Constructing examples of increasingly singular measures with increasingly regular self-convolutions is the topic of several papers (see e.g. [36, 74, 48]). In particular, for any , Körner  constructs a random measure on the line supported on a set of dimension , such that the self-convolution is absolutely continuous and has a Hölder density with exponent , which he shows to be optimal. While Körner has an ad-hoc construction, we show that for our main classes of examples we obtain a similar behaviour, other than for the value of the Hölder exponent. Furthermore, we prove a stronger result: for each , we exhibit a large class of random measures on supported on sets of dimension , such that is absolutely continuous with a Hölder density with exponent , for any which does not have in its spectrum (if is not invertible the conclusion can fail in general, but we still get a weaker result). See Section 13.5.
Recall that a Salem measure is a measure whose Fourier transform decays as fast as its Hausdorff dimension allows, see Section 14.1. We prove that a class of measures, which includes the natural measure on fractal percolation, are Salem measures when their dimension is (and this is sharp), adding to the relatively small number of known examples of Salem measures. See Theorem 14.1. By the result discussed above, these measures have continuous self-convolutions provided they have dimension , and some of them are also Ahlfors regular. The existence of measures with these joint properties is of importance in connection with the restriction problem for fractal measures, and enables us to answer questions of Chen from  and Łaba from , see Section 14.2. We are also able to prove that a wide class of random measures has a power Fourier decay with an explicit exponent, see Corollary 7.5.
1.2. General setup and major classes of examples
Our general setup is as follows. We consider a class of random measures on , obtained as weak limits of absolutely continuous measures with density (we will often identify the densities with the corresponding measures). These are Kahane’s -martingale measures with , together with extra growth and independence conditions to be defined later; intuitively the measure should be thought of as the approximation to at scale (we use dyadic scaling for notational convenience). We pair each with a family of deterministic measures , where the parameter set is a metric space. We study the (mass of the) “intersections” of and with the measures . A priori there is no canonical way to define this, but we employ the fact that is a limit of pointwise defined densities to define the limit of the total masses
provided it exists. Our main abstract result, Theorem 4.1, gives broad conditions on the sequence and the family that guarantee that the function is everywhere well defined and Hölder continuous. These general conditions can be checked in many concrete situations, leading to the consequences and applications outlined above.
To motivate the general results, we describe two key examples, and defer to Section 5 for generalizations and further examples. The first one is fractal percolation, which is also sometimes termed Mandelbrot percolation. Fix an integer and a parameter . We subdivide the unit cube in into equal closed sub-cubes. We retain each of them with probability and discard it with probability , with all the choices independent. For each of the retained cubes, we continue inductively in the same fashion, by further subdividing them into equal sub-cubes, retaining them with probability and discarding them otherwise, with all the choices independent. The fractal percolation limit set is the set of points which are kept at each stage of the construction, see Figure 1 for an illustration of the first few steps of the construction.
It is well known that if , then is a.s. empty, and otherwise a.s. conditioned on . The natural measure on is the weak limit of , where is the union of all retained cubes of side length , and denotes -dimensional Lebesgue measure. Fractal percolation is statistically self-similar with respect to the transformations that map the unit cube to the -adic sub-cubes of size .
The geometric measure theoretical properties of fractal percolation and related models have been studied in depth, see [25, 61, 22, 2, 71, 67, 77, 28]. There is another large class of random sets and measures that has achieved growing interest in the probability literature (see e.g. [14, 65]) but less so in the fractal geometry literature (although the model essentially goes back to Mandelbrot , and Zähle  considered a closely related model). We describe a particular example and leave further discussion to Section 5.1. Let be the measure on , where is a real parameter. Recall that a Poisson point process with intensity is a random countable collection of points such that:
For any Borel set , the random variable is Poisson with mean ( denotes the cardinality of ).
If are pairwise disjoint subsets of , then the random variables are independent.
One can then form the random cut-out set , see Figure 2 for an approximation.
There is a natural measure supported on : it is the weak limit of , where , and , where is a constant depending only on the ambient dimension . It follows from standard techniques that if then almost surely conditioned on ; and otherwise is almost surely empty.
This model is invariant (in law) under arbitrary rotations. In particular, unlike subdivision models in a polyhedral grid, there are no “exceptional directions”. This will be an important feature in some of our applications.
A measure will always refer to a locally finite Borel-regular outer measure on some metric space. The notation stands for the closed ball of centre and radius in a metric space which will always be clear from context. The open ball will be denoted by .
We will use Landau’s and related notation. If is a variable by we mean that there exists such that for all . By we mean , and by we mean that both and hold. As usual, means that . Occasionally we will want to emphasize the dependence of the constants implicit in the notation on other previously defined constants; the latter will be then added as subscripts. For example, means that for some constant which is allowed to depend on .
We always work on some Euclidean space . Most of the time the ambient dimension will be clear from context so no explicit reference will be made to it. We denote by the family of dyadic cubes of with side length . It will be convenient that these are pairwise disjoint, so we consider a suitable half-open dyadic filtration.
The indicator function of a set will be denoted by either or , and we will write for the -neighbourhood . We denote the symmetric difference of two sets by .
We denote the family of finite Borel measures on by . The trivial measure for all sets is considered as an element of . Given , we denote .
As noted earlier, denotes Hausdorff dimension. We denote upper box-counting (or Minkowski) dimension by , and box-counting dimension (when it exists) by . A good introduction to fractal dimensions can be found in [27, Chapters 2 and 3]. For and , we define the lower and upper dimensions of at by
and denote the common value by if they are the same.
Let be the family of isometries of . This is a manifold diffeomorphic to (via ), where is the -dimensional orthogonal group. On and for more general families of linear maps, we use the standard metric induced by the Euclidean operator norm , and this also gives a metric in , and also on the space of affine maps: if for linear and . Likewise, will denote the space of non-singular similarity maps, which is identified with via . The space of contracting similarities, i.e. those for which , will be denoted by . The identity map of is denoted by .
The Grassmannian of -dimensional linear subspaces of will be denoted . It is a compact manifold of dimension , and its metric is
where denotes orthogonal projection. The manifold of -dimensional affine subspaces of will be denoted . It is diffeomorphic to , and this identification defines a natural metric.
The metrics on all these different spaces will be denoted by ; the ambient space will always be clear from context (also note that these metrics and the ambient dimension are denoted by the same symbol ).
For , let be the full shift on symbols. Given , let denote the induced projection map , , and let be the self-similar set . For further background on iterated function systems, including the definitions of the open set and strong separation conditions, see e.g. [26, Section 2.2].
Throughout the paper, , etc, denote deterministic constants whose precise value is of no importance (and their value may change from line to line), while etc. will always denote random real numbers.
We summarize our notation and notational conventions in Table 1. Many of these concepts will be defined later.
|,||Lebesgue measure on ,|
|,||an SI-martingale and its limit|
|Poissonian cutout martingales|
|, ,||specific examples of SI-martingales of various types|
|,||random sets related to an SI-martingale :|
|parametrized family of (deterministic) measures|
|the “intersection” of and .|
|,||total mass of , and its limit|
|the family of compact sets of|
|intensity measures on|
|Poisson point process (with intensity )|
|family of Borel sets consisting of|
|“typical” shapes of a specific SI-martingale|
|an element of|
|the seed of the SI-martingale (the support of )|
|,||the family of real algebraic curves in|
|and the ones of degree at most .|
|the collection of finite Borel measures on .|
|Fourier dimension of|
|,||the family of half-open dyadic cubes of|
|and the ones with side-length|
|the identity map on|
|the family of isometries of|
|the family of affine maps on|
|the manifold of -dimensional affine subspaces of|
|the family of contracting similarities on|
|the self-similar set corresponding to the IFS|
|the code space .|
|open -neighbourhood of a set|
|a quantitative interior of the set|
|regular inner approximation of|
|orthogonal projection onto the linear subspace|
3. The setting
3.1. A class of random measures
In this section we introduce our general setup. Recall that our ultimate goal is to study intersection properties of random measures with a deterministic family of measures (and likewise for their supports). We begin by describing the main properties that will be required of the random measures.
We consider a sequence of functions , corresponding to the densities of absolutely continuous measures (also denoted ) satisfying the following properties:
is a deterministic bounded function with bounded support.
There exists an increasing filtration of -algebras such that is -measurable. Moreover, for all and all ,
There is such that for all and .
There is such that for any -separated family of dyadic cubes of length , the restrictions are independent.
In other words, is a -martingale (with ) in the sense of Kahane  with the extra growth and independence conditions (SI3), (SI4). Intuitively, should be thought of as an absolutely continuous approximation of at scale .
It is well known that a.s. the sequence is weakly convergent; denote the limit by . It follows easily that the sequence is a decreasing sequence of compact sets and that . Note that we do not exclude the possibility that is trivial for some (and hence all sufficiently large) .
We remark that the are actual functions (defined pointwise) and not just elements of . This is crucial because we will be integrating these functions with respect to singular measures. We also note that the fractal percolation and ball cutout examples discussed in the introduction are easily checked to be SI-martingales.
We call condition (SI4) uniform spatial independence. Although it is the central property that sets our class apart from general Kahane martingales, all of our results hold under substantially weaker independence hypotheses. Since many important examples do indeed have uniform spatial independence, in this article we always assume this condition (except for slight variations in Section 13), and defer the study of the weaker conditions and their consequences to a forthcoming article .
Starting with the seminal paper of Kahane , there is a rich literature on -martingales, and the important special case of random multiplicative cascades: see e.g. [52, 12, 10, 11]. In these papers the main emphasis is on the multifractal properties of the limit measures. Our conditions certainly do not exclude multifractal measures (in particular, large classes of random multiplicative cascades are indeed SI-martingales to which many of our results apply), but our emphasis is different, and for simplicity most of our examples will be monofractal measures.
An important special case is that in which for some (possibly random) sequence . Denote . In this case, should be thought of as the approximate value of the Lebesgue volume of , the -neighbourhood of . Recall that the box dimension of a set can be defined as
(See e.g. [27, Proposition 3.2]). Thus, intuitively, should equal . This statement can be verified (for both and ) in many cases, but in the generality of the given hypotheses it may fail.
3.2. Parametrized families of measures
We now introduce the parametrized families of (deterministic) measures. We always assume the parameter space is a totally bounded metric space . We start by introducing some natural classes of examples; we will come back to them repeatedly in the later parts of the paper. In all cases, is a fixed bounded subset of , such as the unit ball.
For some , is the subset of of -planes which intersect , with the induced natural metric, and is -dimensional Hausdorff measure on .
In this example, . Given some , is the family of all algebraic curves of degree at most which intersect , is a natural metric (see Definition 8.4) and is length measure on .
Let be an arbitrary measure, and let be a totally bounded subset of with the induced metric. The measures are . This example generalizes the first one (in which is -dimensional Lebesgue measure on some fixed -plane).
Let , and let be a totally bounded subset of . Suppose that each iterated function system (IFS) satisfies the open set condition. The measure is the natural self-similar measure for the corresponding IFS.
We will occasionally state results for all , where is actually unbounded (for example, ). However in these cases it will be clear that the statement is non-trivial only for those for which intersects a fixed compact set, and this family will be totally bounded.
In most of the examples above, the -mass of small balls is controlled by a power of the radius which is uniform both in the centre and the parameter . This motivates the following definition.
We say that the family has Frostman exponent , if there exists a constant such that
We emphasize that is not unique. In practice, we try to choose as large as possible, but even in that case, is the “worst-case” exponent over all measures, and for some better Frostman exponents may exist.
Our main objects of interest will be the “intersections” of the random measures and with , and their behaviour as varies. Formally, we define:
for each Borel set , and . Note that for each fixed , is again a -martingale, thus there is a.s a weak limit with . We are mainly interested in the asymptotic behaviour of the total mass, and denote
The reason we focus on the masses rather than the actual measures is twofold. Firstly, for some of our target applications, we only want to know that certain fibers containing the support of the are nonempty, and for this suffices. Secondly, itself captures (perhaps surprisingly) detailed information about the measures (and their supports), such as their dimension. See Sections 10–12 for details.
Since our random densities are compactly supported, it follows that a.s. for each fixed , equals . In Theorem 4.1, we prove that in many cases is a.s. defined for all and Hölder continuous with respect to . We call the measures and “intersections”, because our results have corollaries on the size of the intersection of and (see Sections 11 and 12), but also due to the close connection to the more standard intersection measures defined via the slicing method, see [59, Section 13.3]. To emphasize this connection, we include the following proposition (which will not be used later in the paper). We omit the proof, which is a simple exercise combining the definition above with those found in  for the intersections for almost all .
Let and be the translate of a fixed measure under . Let be an SI-martingale. Then, for all , it follows that
In many cases, we can use the results of this paper to show that the above proposition remains true for the limit measures and holds for all , i.e. a.s. for all simultaneously. It is also possible to consider intersections for more general classes of transformations. We do not pursue this direction further since, for our applications, the limit of the total mass is more important (and easier to handle) than the intersection measures , themselves.
The role of uniform spatial independence is to ensure that, with overwhelming probability, the convergence of is very fast, provided does not grow too quickly. This is made precise in the next key technical lemma, which, apart from slight modifications of the same argument, is the only place in the article where spatial independence gets used. Special cases of this appear in ,  and [76, Theorem 3.1], and our proof is similar.
Let be an SI-martingale. Fix , and let such that for all . Write . Then, for any with
it holds that
where is the constant from the definition of SI-martingale.
In particular, this holds uniformly for all measures in a family with Frostman exponent . In the proof we will make use of Hoeffding’s inequality :
Let be zero mean independent random variables satisfying . Then for all ,
Proof of Lemma 3.4.
By replacing by we may assume that . We condition on , and write and for simplicity. The constants implicit in the notation may depend on . We decompose into the families
where is the dyadic cube containing . Then is empty for all (of course it is also empty for all but finitely many other ). For each , let . Then for all , thanks to the martingale assumption (SI2) (recall that we are conditioning on ). Also, by (SI3),
Moreover, since ,
As a first consequence of Lemma 3.4, we deduce that under a natural assumption and, in particular, the limit is non-trivial with positive probability.
Let satisfy for all and some . Let be an SI-martingale such that a.s. for all , where , and suppose that .
Then the sequence converges a.s. to a non-zero random variable . Moreover,
where the implicit constants are independent of but may depend on the remaining data.
Pick . Again write . Then, by Lemma 3.4,
By the Borel-Cantelli lemma, for all but finitely many , so converges a.s. to a random variable . Let . Since is a martingale, for all and therefore, using that , we get that . In particular, if is large enough, then recalling (3.4),
which implies (taking suitably large) that . This gives the first claim.
For the tail bound, we use Lemma 3.4 to conclude that conditional on , we have
Summing over all then gives the claim. ∎
In particular, applying the above to , we obtain that if , then the SI-martingale itself survives with positive probability. The meaning of the hypothesis will be discussed in the next section, after Theorem 4.1.
4. Hölder continuity of intersections
In this section we prove the main abstract result of the paper:
Let be an SI-martingale, and let be a family of measures indexed by a metric space . We assume that there are positive constants such that the following holds:
For any , can be covered by balls of radius for all .
The family has Frostman exponent .
Almost surely, for all and .
Almost surely, there is a random integer , such that
Further, suppose that . Then almost surely converges uniformly in , exponentially fast, to a limit . Moreover, the function is Hölder continuous with exponent , for any
We remark that the special case of this theorem in which is the natural measure on planar fractal percolation, and is the family of length measures on lines making an angle at least with the axes, was essentially proved by Peres and Rams , and we use some of their ideas.
Before presenting the proof, we make some comments on the hypotheses. Condition (H1) says that the parameter space is “almost” finite dimensional. In most cases of interest, can in fact be covered by balls of radius for some fixed (in other words, has finite upper box counting dimension, which clearly implies (H1)).
Hypotheses (H2) and (H3) say that, in some appropriate sense, the deterministic measures have dimension (at least) , and the random measures have dimension (at least) . The hypothesis then says that the sum of these dimensions exceeds the dimension of the ambient space, which is a reasonable assumption if we want these measures to have nontrivial intersection. We will later see that in many examples is a necessary condition even for the existence of , and often even is necessary.
The a priori Hölder condition (H4) may appear rather mysterious: one needs to assume that the functions are Hölder, with a constant that is allowed to increase exponentially in , in order to conclude that the are indeed uniformly Hölder. As we will see, geometric arguments can often be used to establish (H4), making the theorem effective in many situations of interest.
Proof of Theorem 4.1.
Pick constants such that
We observe that such choices are possible because . Also, let
Note that the right-hand side is positive thanks to (4.3). By taking and arbitrarily close to , we can make arbitrarily close to the right-hand side of (4.2). Thus, our task is to show that converges uniformly and exponentially fast to a limit which is Hölder in with exponent .
For each , let be a -dense family with elements, whose existence is guaranteed by (H1).
We first sketch the argument. We want to estimate in terms of , where , and
If , we simply use the a priori Hölder estimate (4.1) to get a deterministic bound. Otherwise, we find in such that and estimate
The term I will be estimated inductively, for II we will use again the a priori estimate (4.1) and to deal with III we appeal to the fact that almost surely, there is such that
for . Since , and , (4.6) follows from the Borel-Cantelli lemma.
If and then, by (4.1),
From now on we consider the case . By definition,
Let be -close to . Pick