Intersections of sets and Fourier analysis

Abstract

A classical theorem due to Mattila (see [7]; see also [10], Chapter 13) says thatfor almost every z ∈ R d , in the sense of Lebesgue measure.In this paper, we replace the Hausdorff dimension on the left hand side of the first inequality above by the lower Minkowski dimension and replace the Lebesgue measure of the set of translates by a Hausdorff measure on a set of sufficiently large dimension. Interesting arithmetic issues arise in the consideration of sharpness examples. These results are partly motivated by those in [1] and [6] where in the former the classical regular value theorem from differential geometry was investigated in a fractal setting, and in the latter discrete incidence theory is explored from an analytic standpoint.Institute des HautesÉtudes Scientifiques, 35 route des Chartres,

Hausdorff dimension and projections related to intersections

“…Hence the exact analogues of the intersection results of this paper are false even when one of the sets is a plane. However, Järvenpää in [19] and [20] and Eswarathasan, Iosevich, and Taylor in [8] proved some related results.…”

Finite Point Configurations and the Regular Value Theorem in a Fractal setting

“…x pEqq ă τ uq ď maxpΓ k pτ, dim H pEqq, 2 ´dim H pEqq ` . First, consider the range dim H pEq ď 5 4 and fix τ P p0, 4pk´1q 3 dim H pEq `5´2k 3 q. Fix ą 0, it suffices to show that given any set F Ă R 2 satisfying (3.22) dim H pF q ą maxpΓ k pτ, dim H pEqq, 2 ´dim H pEqq ` , there exists a point x P F such that dim H pS k x pEqq ě τ .…”

“…In the continuous case, similar questions for distance sets have been studied in [4,5,24], and it was observed that one needs to assume some minimal regularity for the set E in order for the question to be meaningful (see [24, Page 253] for a discussion). We will work with sets that are "tδ i u-discrete α-regular" below, which generalizes a concept that was first introduced in [19] and used in [24] for the distance set (k " 1) case.…”

“…a pair of compact sets with positive α-dimensional Hausdorff measure for some α ą 5 4 . Further, suppose that there exist Borel probability measures µ 1 and µ 2 on E 1 and E 2 respectively which satisfy µ i pBpx, rqq À r α for i " 1, 2.…”

An elementary approach to simplexes in thin subsets of Euclidean space

“…But the uniform estimate in (1.2) is considerably stronger than the positivity of the Lebesgue measure. For example, in [10], the authors proved that, in the case k = d = 2, (1.2) holds if dim H (E) > 7 4 , yielding not only the continuous Falconer-type configuration result but also a discrete result: If A ⊂ R 2 is a finite homogeneous set with |A| = N, then the number of triples of points from A determining an equilateral triangle of fixed side length does not exceed CN 9 7 , an improvement over the previously known Cn 3 bound (which is a consequence of the Szemeredi-Trotter incidence theorem). (For applications of continuous incidence bounds in geometric measure theory, see, e.g., [7]; for the definition of homogeneous set, see [17].…”