SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2208-2220, January 2020.
For a subset $I\subseteq \mathbb{F}_{q}^{n}$, let $\Delta(I)$ be the set of distances determined by the elements of $I.$ The Erdös--Falconer distance problem in $\mathbb{F}_{q}^{n}$ asks for a threshold on the cardinality $|I|$ so that $\Delta(I)$ contains a positive proportion of the whole distance set. In this paper, we consider the analogous question under Hamming distance, which is the most important metric in coding theory. When $q\geqslant 4$ is a fixed prime power and $n$ goes to infinity, our main result shows that, for arbitrary positive proportion $\alpha,$ we can find $\alpha n$ distinct Hamming distances in $\Delta(I)$ if $|I|>q^{(1-\beta)\cdot n},$ where $\beta$ is a positive number depending on $\alpha.$ Unlike using Fourier analytical method as usual, our main tools include the celebrated dependent random choice and some results from additive number theory and coding theory. Hence our bound is much smaller than the previously known bound which was obtained by Fourier analytic machinery.

中文翻译：

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有限域向量空间上汉明度量下的Erdös-Falconer距离问题

SIAM离散数学杂志，第34卷，第4期，第2208-2220页，2020年1月。
对于子集$ I \ subseteq \ mathbb {F} _ {q} ^ {n} $，令$ \ Delta（I）$为由$ I元素确定的距离集。$Erdös-Falconer距离$ \ mathbb {F} _ {q} ^ {n} $中的问题要求基数$ | I | $的阈值，以便$ \ Delta（I）$包含整个距离集中的正比例。在本文中，我们考虑了汉明距离下的类似问题，这是编码理论中最重要的指标。当$ q \ geqslant 4 $是固定的质数幂并且$ n $变为无穷大时，我们的主要结果表明，对于任意正比例$ \ alpha，$我们可以找到$ \ alpha n $在$ \ Delta中不同的汉明距离（I）$如果$ | I |> q ^ {（1- \ beta）\ cdot n}，$其中$ \ beta $是一个正数，取决于$ \ alpha。$与通常使用傅立叶分析方法不同，我们的主要工具包括著名的依存随机选择以及可加数理论和编码理论的一些结果。因此，我们的界限比通过傅立叶分析机器获得的先前已知的界限小得多。