On Hausdorff dimensions of $k$-point configuration sets and Elekes-R\'onyai type theorems

TL;DR

This paper extends the Elekes-Rónyai theorem to trivariate real analytic functions, showing that such functions either have a specific form or significantly increase the Hausdorff dimension of images of sets, with implications for geometric measure theory.

Contribution

It proves a dimension expansion version of the Elekes-Rónyai theorem for trivariate functions and improves results for bivariate functions, using advanced Fourier analysis techniques.

Findings

Dimension expansion for trivariate functions when the set dimension is between 1/2 and 1.

If the set dimension exceeds 2/3, the image has positive Lebesgue measure.

Generalization of Falconer-type results for configuration sets with positive measure.

Abstract

We prove a ''dimension expansion'' version of the Elekes-R\'onyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is true: whenever a Borel set $A\subset\mathbb{R}$ has Hausdorff dimension $\alpha\in \left(\frac{1}{2},1\right)$, $f(A\times A\times A)$ has dimension significantly larger than that of $A$, i.e. \begin{align*} \dim_Hf(A\times A\times A)\geq \alpha+\varepsilon(\alpha),\quad \text{for some } \varepsilon(\alpha)>0, \end{align*} Moreover, if $\alpha>\frac{2}{3}$, $f(A\times A\times A)$ has positive Lebesgue measure. This is a considerable extension of the result established by Koh, T. Pham, and Shen (J. Funct. Anal. 286 (2024)). We also obtain an alternative proof and an improvement for the Elekes-R\'onyai type theorem for bivariate real analytic functions established by Raz and Zahl (Geom. Funct. Anal. 34 (2024)). We derive these from more general results, showing that various $k$-point configuration sets of thin sets have positive Lebesgue measure by exploiting the optimal $L^2$-based Sobolev estimates for the associated family of Fourier integral operators. Extending the framework developed by Greenleaf, Iosevich, and Taylor (Mathematika 68 (2022), Math. Z. 306 (2024)) to prove Mattila-Sj\"olin type theorems, we obtain Falconer-type results for many configuration sets on which the method would be vacuous if demanding nonempty interior. In particular, when $k=2$, we generalize the Falconer-type result for metric functions in $\mathbb{R}^d$ satisfying strong non-vanishing curvature conditions established by Eswarathasan, Iosevich, and Taylor (Adv. Math. 228 (2011)) and the asymmetric Mattila-Sj\"olin type results of Greenleaf, Iosevich, and Taylor (J. Geom. Anal. 31 (2021)) to a broader class of smooth functions of asymmetric form.