Intersection patterns and connections to distance problems
Thang Pham, Semin Yoo
TL;DR
The paper develops a finite-field framework for intersection problems under rigid motions, establishing that for almost every rotation g, the cross-intersection |A ∩ (g(B)+z)| attains near-typical size ~ |A||B|/q^d for many shifts z, which in turn implies strong expansion of A − gB. The authors deploy a two-pronged approach combining incidence geometry for points and rigid motions with Fourier-analytic L^2 distance bounds, leveraging restriction theory and modern incidence bounds to obtain dimension-uniform results and sharpness in odd dimensions. In the planar prime-field setting, they derive a prime-field Rotational Erdős–Falconer distance theorem and provide explicit growth and intersection bounds, linking intersection phenomena with distance problems across dimensions. Taken together, the results create a robust two-way bridge between intersection patterns and distance problems over finite fields, with quantitative, dimension-uniform conclusions and several sharp thresholds. The work has notable implications for discrete harmonic analysis and geometric combinatorics in finite-field models of distance and incidence questions.
Abstract
Let $A$ and $B$ be sets in a finite vector space. In this paper, we study the magnitude of the set $A\cap f(B)$, where $f$ runs through a set of transformations. More precisely, we will focus on the cases that the set of transformations is given by orthogonal matrices or orthogonal projections. We prove that if $A, B\subset \mathbb{F}_q^d$ satisfy some natural conditions, then, for almost every $g\in O(d)$, there are at least $\gg q^d$ elements $z\in \mathbb{F}_q^d$ such that \[|A\cap (g(B)+z)| \sim \frac{|A||B|}{q^d}.\] This implies that $|A-gB|\gg q^d$ for almost every $g\in O(d)$. In the flavor of expanding functions, with $|A|\le |B|$, we also show that the image $A-gB$ grows exponentially. In two dimensions, the result simply says that if $|A|=q^x$ and $|B|=q^y$, as long as $0<x\le y<2$, then for almost every $g\in O(2)$, we can always find $ε=ε(x, y)>0$ such that $|A-gB|\gg |B|^{1+ε}$. To prove these results, we need to develop new and robust incidence bounds between points and rigid motions by using a number of techniques including algebraic methods and discrete Fourier analysis. Our results are essentially sharp in odd dimensions. In the prime field plane, we further employ recent $L^2$ distance bounds and point-line/plane incidence machinery to derive improvements. Notable applications include a strong prime field analogue of a question of Mattila related to the Falconer distance problem, the Rotational Erdős-Falconer distance problem, and a quadratic expansion law. Taken together, the results in this paper present a robust two-way link between intersection phenomena and distance problems over finite fields, with dimension-uniform consequences and sharpness in several ranges.