Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting

TL;DR

This paper develops multi-parameter projection theorems for fractal sets and applies them to sum-product problems and finite point configurations, revealing size and measure properties in Euclidean spaces.

Contribution

It introduces new projection estimates for fractal sets and demonstrates their applications to sum-product phenomena and the measure of point configurations.

Findings

Size estimates for sum-product sets based on Hausdorff dimension

Positive Lebesgue measure results for sets of k-simplices with large Hausdorff dimension

Connections established between projection theorems and number theoretic estimates

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 $A \cdot A+...+A \cdot A$, where $A$ is a subset of the real line of a given Hausdorff dimension, $A+A=\{a+a': a,a' \in A \}$ and $A \cdot A=\{a \cdot a': a,a' \in A\}$. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ${\Bbb R}^d$ is sufficiently large, then the ${k+1 \choose 2}$-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.