1,721,009 research outputs found
Absolute continuity of complex Bernoulli convolutions
Get PDFWe 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.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Solomyak, Boris. Bar Ilan University; Israe
On the Hausdorff dimension of pinned distance sets
Get PDFWe 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.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
On packing measures and a theorem of Besicovitch
Get PDFLet 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.Fil: Garcia, Ignacio Andres. Universidad Nacional de Mar del Plata; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Mar del Plata. Instituto de Investigaciones Físicas de Mar del Plata; ArgentinaFil: Shmerkin, Pablo. University Of Surrey; Reino Unid
Salem Sets with No Arithmetic Progressions
Get PDFWe 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 pseudo-random subsets of R which do contain progressions.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
Get PDFWe prove that the set of exceptional λ∈(1/2,1)λ∈(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.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
Improved bounds for the dimensions of planar distance sets
Get PDFWe 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 2 A such that the pinned distance set 1jx -yjW x 2 Aºhas 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.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. University of British Columbia; Canad
Sets which are not tube null and intersection properties of random measures
Get PDFWe 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.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Suomala, Ville. Universidad de Oulu; Finlandi
Absolute continuity of self-similar measures, their projections and convolutions
Get PDFWe 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.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaFil: Solomyak, Boris. University of Washington; Estados Unido
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
Get PDFWe study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the Lq dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of ×p- and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an Lq density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to Lq norms, and likewise relies on an inverse theorem for the decay of Lq norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentin
Hölder coverings of sets of small dimension
No full textWe show that a set of small box counting dimension can be covered by a H¨older 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, H¨older graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on H¨older coverings in polar coordinates and, on the other hand, prove that a Homogenous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions.Fil: Rossi, Eino Vihtori. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentin
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