Falconer-type estimates for dot products

TL;DR

This paper constructs sharpness examples for Falconer-type estimates involving dot products, demonstrating the precise dimensional threshold where such estimates hold or fail, extending previous results from Euclidean distances to dot products.

Contribution

It provides new sharpness examples for Falconer-type dot product estimates, establishing the exact dimensional threshold for their validity and generalizing prior Euclidean distance results.

Findings

Constructed measures with finite energy but failing dot product estimates

Proved the sharpness of the dimensional threshold at s=(d+1)/2

Extended results from Euclidean distances to dot products and convex norms

Abstract

We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $\mu$ satisfying the energy estimate $I_s(\mu)<\infty,$ yet the estimate \begin{equation} (\mu \times \mu)\{(x,y):1\leq x\cdot y \leq 1+\epsilon\} \leq C\epsilon \end{equation} does not hold with constants independent of $\epsilon$. It is known (\cite{EIT11}) that such an estimate always holds with $C$ independent of $\epsilon$ if $I_{\frac{d+1}{2}}(\mu)<\infty$. Thus our estimate proves the sharpness of the dimensional threshold in this result and generalizes similar results (\cite{Mat95}, \cite{IS16}) established in the case when the dot product $x \cdot y$ is replaced by the Euclidean distance function $|x-y|$, or, more generally, ${||x-y||}_K$, the distance that comes from the norm induced by a symmetric convex body $K$ with a smooth boundary and non-vanishing curvature. Our constructions are partially based on ideas that come from discrete incidence theory.