On Falconer type functions and the distance set problem

TL;DR

This work develops a projection-based framework for Falconer-type functions $\Phi:\mathbb{R}^m\times\mathbb{R}^p\to\mathbb{R}$, proving that large Hausdorff dimensions of compact sets $A,B$ force the image $\Delta_\Phi(A,B)$ to have positive Lebesgue measure under mild regularity. It reinterprets $\Phi$ as a projection of a vector-valued map, enabling application of classical projection results to broad classes of smooth functions, including certain higher-degree polynomials. The authors obtain a versatile general theorem for $\Phi(x,y)=\sum P_k(x)Q_k(y)$ and derive a polynomial nonhomogeneity criterion, extending Koh-Pham-Shen's results to more general multivariate polynomials. As concrete applications, they prove new positive results for Falconer’s distance problem in regimes where previous bounds fail, notably for pins on a hyperplane and for certain Cartesian product sets (e.g., with a Salem component), yielding sharp dimensional thresholds and corroborating conjectures in these structured cases. The paper also furnishes sharp examples illustrating the necessity of assumptions and delineates the scope and limitations of the projection approach.

Abstract

Let $Φ: \mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}$ be a smooth function, and let $A\subset\mathbb{R}^n$ and $B\subset \mathbb{R}^m$ be compact sets, with $n,m\geq 1$. We show that if $Φ$ satisfies suitable regularity conditions and the Hausdorff dimensions of $A$ and $B$ are sufficiently large, then the image \begin{align*} Δ_Φ(A,B):=\{ Φ(x,y): x\in A, y\in B\} \end{align*} has positive Lebesgue measure. Our projection theoretic approach provides a versatile framework applicable to a broad class of smooth functions of interest. This result not only recovers the sharp theorem of Koh-Pham-Shen \cite{Koh2024} for quadratic polynomials in three variables, but also extends to certain classes of higher degree multivariate polynomials. %The key feature of our argument is the interpretation of the original map $Φ$ as a suitable projection, allowing the use of well known projection theorems. As a notable application, we establish new positive results for the Falconer's distance problem in certain regimes, particularly where the best known bounds fail to apply. Specifically, if $A, B\subset \mathbb{R}^n$ are compact sets, $n\geq 2$, $B$ is contained in a hyperplane, and \begin{align*} \dim_{\mathcal{H}} A+\dim_{\mathcal{H}} B>n, \end{align*} then the distance set $ Δ(A,B):=\{ \vert x-y\vert: x\in A, y\in B\}$ has positive Lebesgue measure; moreover, the dimensional threshold is sharp. We further show that the Falconer's distance conjecture holds for a class of product sets under additional assumptions on their Fourier dimensions. Namely, if $A=A_1\times A_2\subset \mathbb{R}^{n-1}\times \mathbb{R}$, where $A_2$ is a Salem set, and \begin{align*} \dim_\mathcal{H}A>\frac{n}{2}, \end{align*} then the distance set $Δ(A):=\{ \vert x-y\vert: x,y\in A\}$ has positive Lebesgue measure.

Figures (1)

• Figure 1: Dimensional thresholds for the distance set problem for Cartesian product sets $A = A_1 \times \cdots \times A_n$, where $\alpha =\min\limits_{i=1,\ldots,n} \dim_{\mathcal{H}} A_i$, $\gamma_2=\tfrac{5}{4}$, and $\gamma_n=\tfrac{n}{2}+\tfrac{1}{4}-\tfrac{1}{8n+4},\quad \forall n\geq 3$.

Theorems & Definitions (26)

• Conjecture
• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.1
• Theorem 1.4
• Remark 1.5
• Corollary 1.2
• Remark 1.6
• Theorem 1.7