Intersection patterns and connections to distance problems

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 $\epsilon=\epsilon(x, y)>0$ such that $|A-gB|\gg |B|^{1+\epsilon}$. 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 Erd\H{o}s-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.