On Falconer type functions and the distance set problem

Abstract

We study the distance set problem for pairs of compact sets $A, B\subset \mathbb{R}^n$, $n\geq 2$. We show that if $B$ is contained in a hyperplane and \begin{align*} \dim_{H} A+\dim_{H} B>n, \end{align*} then the distance set $ \Delta(A,B):=\left\{ \vert x-y\vert: x\in A, y\in B\right\}$ has positive Lebesgue measure, and the dimensional threshold is sharp. This yields new positive results for Falconer's distance problem in certain regimes, particularly where the best known bounds fail to apply. We further establish Falconer's distance conjecture for certain classes of product sets under additional structural assumptions. Specifically, if $A=A_1\times A_2\subset \mathbb{R}^{m}\times \mathbb{R}^{n-m}$ for some $0\leq m\leq n-1$, where $A_2$ is a Salem set, and \[ \dim_HA>\frac{n}{2}, \] then the distance set $\Delta(A):=\left\{|x-y|: x,y\in A\right\} $ has positive Lebesgue measure. A key feature of our argument is the interpretation of the original map as a suitable projection. We extend the analysis to a broad class of smooth functions, recovering the sharp result of Koh, Pham, and Shen (J. Funct. Anal. 286 (2024)) for quadratic polynomials in three variables.