Pinned Dot Product Set Estimates

TL;DR

This paper investigates the size and structure of dot product sets derived from fractal subsets of Euclidean space, providing new bounds and insights into their Hausdorff dimension and measure.

Contribution

It establishes new lower bounds on the Hausdorff dimension needed for the dot product sets to be large, extending previous results in projection and distance set theory.

Findings

Lower bounds on Hausdorff dimension for large dot product sets

Results on positive measure and nonempty interior of these sets

Application of classical and recent projection theory techniques

Abstract

We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y= \alpha, \text{ for some $y\in A$}\}. \] We discuss some of what is already known to give a picture of the current state of the art, as well as prove some new results and special cases. We obtain lower bounds on the Hausdorff dimension of $A$ to guarantee that $\Pi^a_x(A)$ is large in some quantitative sense for some $a\in A$ (i.e. $\Pi_x^a(A)$ has large Hausdorff dimension, positive measure, or nonempty interior). Our approach to all three senses of "size" is the same, and we make use of both classical and recent results on projection theory.