Multiparameter Projection Theorems with Applications to Sums-Products and Finite Point Configurations in the Euclidean Setting

Abstract

In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of A · A + · · ·+ A · A, where A is a subset of the real line of a given Hausdorff dimension, A+A = {a+a ′ : a, a ′ ∈ A} and A · A = {a · a ′ : a, a ′ ∈ A}. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of R d is sufficiently large, then the k+1 2 -dimensional Lebesgue measure of the set of k-simplexes determined by this set is positive. The sharpness of these results and connection with number theoretic estimates is also discussed.

A group-theoretic viewpoint on Erdős–Falconer problems and the Mattila integral

“…Remark 1.7. While the results of Theorem 1.5 significantly improve and extend the exponents in [4,9,8], the group-theoretic nature of our methods also casts new light upon the classical Mattila integral (see Sec. 5), potentially leading to further progress on related problems.…”

“…A natural extension of the Falconer distance problem is the congruent simplex problem [4,9,8]. We say that {x 1 , .…”

Finite configurations in sparse sets

“…For extensions of Theorem 1.8 to finite configurations in higher dimensions, see e.g. [6], [11], [14].…”

“…forms a basis for R6 . It follows that if a, b, c ∈ E, there is a unique y ∈ R 6 such that b = a + B 2 y and c = a + B 3 y.…”

On an Application of Guth–Katz Theorem

“…In the continuous setting, both problems are studied in [7]. For finite field versions of these problems see, for example, [12] and [2].…”