Finite configurations in sparse sets

Abstract

Let E ⊆ R n be a closed set of Hausdorff dimension α. For m ≥ n, let {B 1 , . . . , B k } be n × (m − n) matrices. We prove that if the system of matrices B j is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B 1 y, . . . , B k y}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in R n and isosceles right triangles in R 2 ). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of R.

Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions

“…Proof of Theorem 2.3 (2). To prove the required statement, we will show that lim n→∞ | µ(N 1 ...N n )| > 0.…”

Large sets avoiding patterns

“…We now proceed to define the subcollection I ′ N (J, v) and the set S v that obey the requirements in (3). Consider the collection…”

“…A counterpoint to [10] is a result of Laba and the second author [12], who have established existence of three-term progressions in special "random-like" subsets of R that support measures satisfying an appropriate ball condition and a Fourier decay estimate. Higher dimensional variants of this theme may be found in [3,9]. On the other hand, large Hausdorff dimensionality, while failing to ensure specific patterns, is sometimes sufficient to ensure existence or even abundance of certain configuration classes; see for instance the work of Iosevich et al [5,6,2,7].…”

Configuration Sets with Nonempty Interior

“…Y is a smooth surface of codimension k. As in the paper [10] by Grafakos, Palsson and the first two authors, for a compact subset E ⊂ R d , we define the (two-point) Φ-configuration set of E as Our goal is to find a threshold, s 0 = s 0 (Φ), such that if dim H (E) > s 0 then ∆ Φ (E) has nonempty interior. Similar questions have been studied in, or are accessible to the methods of, a number of works, e.g., [11,2,5,23,24,21]. (There is of course an extensive literature on the related Falconer distance problem [8], and its generalizations to configurations, where the question is what lower bound on dim H (E) ensures that ∆ Φ (E) has positive Lebesque measure [41,7,6,15].)…”