The Mattila-Sj\"{o}lin problem for the $k$-distance over a finite field

다국어 초록 (Multilingual Abstract)

Let $\Fq{d}$ be a $d$-dimensional vector space over a finite field $\Fq{}$ with $q$ elements. For $x\in \Fq{d}$, let $\knorm{}{x} = x_1^2+\cdots +x_d^2$. By abuse of terminology, we shall call $\knorm{}{\cdot}$ a norm on $\Fq{d}$. For a subset $E\subset \Fq{d}$, let $\Delta(E)$ be the distance set on $E$ defined as \[\Delta(E):=\{\knorm{}{x-y} : x, y \in E \}.\] The Mattila-Sj\"{o}lin problem seeks the smallest exponent $\alpha>0$ such that $ \Delta(E) =\Fq{}$ for all subsets $E \subset \Fq{d}$ with $|E| \geq Cq^\alpha$. In this article, we consider this problem for a variant of this norm, which generates a smaller distance set than the norm $\knorm{}{\cdot}.$ Namely, we replace the norm $\knorm{}{\cdot}$ by the so-called $k$-norm $(1 \leq k \leq d)$, which can be viewed as a kind of deformation of $\knorm{}{\cdot}$. To derive our result on the Mattila-Sj\"{o}lin problem for the $k$-norm, we use a combinatorial method to analyze various summations arising from the discrete Fourier machinery. Even though our distance set is smaller than the one in the Mattila-Sj\"{o}lin problem, for some $k$ we still obtain the same result as that of Iosevich and Rudnev \cite{IR07}, which deals with the Mattila-Sj\"{o}lin problem. Furthermore, our result is sharp in all odd dimensions.