Multi-parameter 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 $A\cdot A+...+A\cdot A$, where $A$ is a subset of the real line of a given Hausdorff dimension, $A+A=\{a+a^{\prime }:a,a^{\prime }\in A\}$ and $A\cdot A=\{a\cdot a^{\prime }:a,a^{\prime }\in A\}$. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ${\mathbb{R}}^d$ is sufficiently large, then the $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$-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.