Search

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let be non-empty Borel sets. If X is not contained in any line, we prove that If dimHY>1, we have the following improved lower bound: Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines , then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}. While the results above concern , we also derive some counterparts in by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren. [ABSTRACT FROM AUTHOR]

We prove a version of Bourgain's projection theorem for parametrized families of C2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in R² or close to 3=2 in R³, then for y ∈ A outside of a very sparse set, the pinned distance set {|x - y|: x ∈ A} has Hausdorff dimension at least 1=2 C c, where c is universal. Furthermore, the same holds if the distances are taken with respect to a C² norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between ı-balls and δ-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into "Frostman pieces" that may be of independent interest. [ABSTRACT FROM AUTHOR]

We prove that if α ∈ (0, 1/2], then the packing dimension of a set E ⊂ ℝ2 for which there exists a set of lines of dimension 1 intersecting E in Hausdorff dimension ≥ α is at least 1/2 + α + c(α)for some c(α) > 0. In particular, this holds for α-Furstenberg sets, that is, sets having intersection of Hausdorff dimension ≥ ≥ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of α-Furstenberg sets over the "trivial" estimate for all values of α ∈ (0, 1). The proof extends to more general families of lines, and shows that the scales at which an α-Furstenberg set resembles a set of dimension close to 1/2 + α, if they exist, are rather sparse. [ABSTRACT FROM AUTHOR]

We show that given α∈(0,1) there is a constant c=c(α)>0 such that any planar (α,2α)-Furstenberg set has Hausdorff dimension at least 2α+c. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps. [ABSTRACT FROM AUTHOR]

We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ Rn is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian. [ABSTRACT FROM AUTHOR]

We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid ε -approximations of arithmetic progressions. Some of these estimates are in terms of Szemerédi bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN 14:4419–4430, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent to box or Assouad dimension for this problem, and obtain a lower bound for Fourier dimension. [ABSTRACT FROM AUTHOR]

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if dimH (A) > 1, then the set of distances spanned by points of A has Hausdorff dimension at least 40/57 > 0.7 and there are many y ∈ A such that the pinned distance set {\x - y\: x ∈ A} has Hausdorff dimension at least 29/42 and lower box-counting dimension at least 40/57. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input. [ABSTRACT FROM AUTHOR]

The Lq dimensions, for 1 < q < 8, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the Lq dimension improve under convolution? This can be seen as a variant of the well-known Lp-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the Lq dimension. We also study the case q = 8, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the Lq norms of convolutions due to the second author. [ABSTRACT FROM AUTHOR]

We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer'edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive -measure, where is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well-known features of independent percolation on trees to a setting with long-range dependencies. [ABSTRACT FROM AUTHOR]

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R 2 is a Borel set of Hausdorff dimension s > 1 , then its distance set has Hausdorff dimension at least 37 / 54 ≈ 0.685 . Moreover, if s ∈ (1 , 3 / 2 ] , then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set { | x - y | : x ∈ A } has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1 + s + 3 s (2 - s)) ≥ 0.933 . These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension > 1 . Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions. [ABSTRACT FROM AUTHOR]

We show that a set of small box counting dimension can be covered by a Hölder graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We observe that, as a consequence, Hölder graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on Hölder coverings in polar coordinates and, on the other hand, prove that a homogeneous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions. [ABSTRACT FROM AUTHOR]

We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension 1 (in particular, for many x ∈ A). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint s = 1. [ABSTRACT FROM AUTHOR]

Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of B'ar'any, Hochman- Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions. [ABSTRACT FROM AUTHOR]

We study the relationship between the size of two sets B, S ⊂ R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others. [ABSTRACT FROM AUTHOR]

We construct compact Salem sets in R/Z of any dimension (including 1), which do not contain any arithmetic progressions of length 3. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than 1, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension 1. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions) and helps clarify a result of Łaba and Pramanik on pseudorandom subsets of R which do contain progressions. [ABSTRACT FROM AUTHOR]

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures. [ABSTRACT FROM PUBLISHER]

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results. [ABSTRACT FROM PUBLISHER]

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small--of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand's projection theorem for planar self-similar sets. [ABSTRACT FROM AUTHOR]

We give a fractal-geometric condition for a measure on $$[0,1]$$ to be supported on points $$x$$ that are normal in base $$n$$ , i.e. such that $$\{n^kx\}_{k\in \mathbb {N}}$$ equidistributes modulo 1. This condition is robust under $$C^1$$ coordinate changes, and it applies also when $$n$$ is a Pisot number rather than an integer. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations. [ABSTRACT FROM AUTHOR]

We employ the ergodic-theoretic machinery of scenery flows to address classical geometric measure-theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its settheoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases. [ABSTRACT FROM AUTHOR]

We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution (FD) in the sense of Hochman is generated by a USM, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery flow. We further show that the collection of FDs is closed under the weak topology and, moreover, is a Poulsen simplex, that is, extremal points are dense. We apply these to show that a Baire generic measure is as far as possible from being uniformly scaling: at almost all points, it has all FDs as tangent distributions. [ABSTRACT FROM AUTHOR]

We show that in Rd there are purely unrectifiable sets of Hausdorff (and even box counting) dimension d - 1 which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same authors and by Carbery. Our method extends also to 'convex tube null sets', establishing a contrast with a theorem of Alberti, Csörnyei and Preiss on Lipschitz-null sets. The sets we construct are random, and the proofs depend on intersection properties of certain random fractal measures with curves. [ABSTRACT FROM AUTHOR]

The pressure function P( A, s) plays a fundamental role in the calculation of the dimension of 'typical' self-affine sets, where A = ( A, ..., A) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of 'typical' self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles. [ABSTRACT FROM AUTHOR]

Let Hh be the h-dimensional Hausdorff measure on Rd. Besicovitch showed that if a set E is null for Hh, then it is null for Hg, for some dimension g smaller than h. We prove that this is not true for packing measures. Moreover, we consider the corresponding questions for sets of non-σ-finite packing measure and for pre-packing measure instead of packing measure. [ABSTRACT FROM AUTHOR]

We prove that the set of exceptional $${\lambda\in (1/2,1)}$$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform. [ABSTRACT FROM AUTHOR]

We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize and simplify a number of recent results on local homogeneity, porosity and conical densities of measures. [ABSTRACT FROM PUBLISHER]

We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are one dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts. [ABSTRACT FROM AUTHOR]

Using probabilistic ideas, we prove that if μ is a mean porous measure on , then the packing dimension of μ is strictly smaller than n. Moreover, we give an explicit bound for the packing dimension, which is asymptotically sharp in the case of small porosity. This result was stated in [D. B. Beliaev and S. K. Smirnov, “On dimension of porous measures”, Mathematische Annalen 323 (2002): 123–41], but the proof given there is not correct. We also give estimates on the dimension of weakly mean porous measures, which improve another result of Beliaev and Smirnov. [ABSTRACT FROM PUBLISHER]

Denote by µ the distribution of the random sum $$(1 - a)\sum\nolimits_{j = 0}^\infty {{w_j}{a^j}} $$, where P( ω = 0) = P( ω = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ is supported on C, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 − 2 a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ* (µ ∘ S), where S( x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ ≠ 0, then , where D denotes any of correlation, Hausdorff or packing dimension of a measure. We also show that, perhaps surprisingly, for uncountably many values of λ the convolution µ* (µ ∘ S) is a singular measure, although dim( C) + dim( C) > 1 and log(1/3)/ log(1/4) is irrational. [ABSTRACT FROM AUTHOR]

Let νpλ be the distribution of the random series $\sum_{n=1}^\infty i_n \lam^n$, where in is a sequence of i.i.d. random variables taking the values 0, 1 with probabilities p, 1 − p. These measures are the well-known (biased) Bernoulli convolutions.In this paper we study the multifractal spectrum of νpλ for typical λ. Namely, we investigate the size of the sets \begin{linenomath} \[ \Delta_{\lam, p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log \nula^p(B(x, r))}{\log r} =\alpha\right\}\!. \]\end{linenomath} Our main results highlight the fact that for almost all, and in some cases all, λ in an appropriate range, Δλ, p(α) is nonempty and, moreover, has positive Hausdorff dimension, for many values of α. This happens even in parameter regions for which νpλ is typically absolutely continuous. [ABSTRACT FROM PUBLISHER]

We compute the Minkowski dimension for a family of self-affine sets on ℝ². Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets. [ABSTRACT FROM AUTHOR]