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 ℝd 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. © © Springer Science+Business Media, LLC 2013.