Intersection patterns and connections to distance problems
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914171511111680 |
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| author | Pham, Thang Yoo, Semin |
| author_facet | Pham, Thang Yoo, Semin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_08004 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intersection patterns and connections to distance problems Pham, Thang Yoo, Semin Combinatorics Classical Analysis and ODEs Number Theory 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. |
| title | Intersection patterns and connections to distance problems |
| topic | Combinatorics Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2304.08004 |