Falconer type functions in three variables

TL;DR

This paper proves that for certain quadratic functions in three variables, the image of three sets with sufficiently large Hausdorff dimension has positive Lebesgue measure, extending Falconer-type results.

Contribution

It establishes a new Falconer-type result for quadratic functions in three variables, combining harmonic analysis and finite field combinatorics.

Findings

Image set has positive Lebesgue measure when sum of Hausdorff dimensions exceeds 2.

Utilizes a harmonic analysis result by Eswarathasan, Iosevich, and Taylor.

Employs a combinatorial argument from finite field models.

Abstract

Let $f\in \mathbb{R}[x, y, z]$ be a quadratic polynomial that depends on each variable and that does not have the form $g(h(x)+k(y)+l(z))$. Let $A, B, C$ be compact sets in $\mathbb{R}$. Suppose that $\dim_H(A)+\dim_H(B)+\dim_H(C)>2$, then we prove that the image set $f(A, B, C)$ is of positive Lebesgue measure. Our proof is based on a result due to Eswarathasan, Iosevich, and Taylor (Advances in Mathematics, 2011), and a combinatorial argument from the finite field model.