2023/04/17 by Pham, Thang, Yoo, Semin · 3 citations
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2304.08004
Let A and B be sets in a finite vector space. In this paper, we study the magnitude of the set A∩ 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⊂ \mathbbFqd satisfy some natural conditions, then, for almost every g∈ O(d), there are at least ≫ qd elements z∈ \mathbbFqd such that |A∩ (g(B)+z)| ∼ (|A||B|)/(qd). This implies that |A-gB|≫ qd for almost every g∈ O(d). In the flavor of expanding functions, with |A|≤ |B|, we also show that the image A-gB grows exponentially. In two dimensions, the result simply says that if |A|=qx and |B|=qy, as long as 00 such that |A-gB|≫ |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 L2 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.