发表机构
Virginia Tech（弗吉尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出整数格中的Erdős-Falconer距离问题，以离散类比、Falconer问题等为动机，用离散球面平均的ℓᵖ改进不等式研究，其方法在部分场景优于现有结果，或可助力原始固定距离问题。
AI 中文摘要
本文提出并研究了所谓的整数格中的Erdős-Falconer距离问题。一个动机来自Magyar对Furstenberg、Katznelson和Weiss结果的离散类比，后者证明了对于ℝᵈ（d≥2）中具有正上Banach密度的集合，必须出现所有任意大的距离。另一个动机是Falconer距离问题，该问题可视为对Steinhaus结果的改进，将假设从正Lebesgue测度放宽到仅询问集合的维数。在这一背景下，我们希望降低正上Banach密度的假设，同时获得保证存在大量不同距离的结论。所有这些问题也与Erdős不同距离问题相关。我们的命名方式与该问题的有限域变体一致，承认该问题的部分内容与Erdős不同距离问题相似，而另一部分则与Falconer距离问题相似。我们使用的主要工具是离散球面平均的ℓᵖ改进不等式，据我们所知，这是此类界的首次应用。可以利用Erdős不同距离问题的结果来推进我们的猜想，但令人惊讶的是，即使使用Solymosi和Vu的机制在高维中最强的可用界，在某些情况下我们的方法表现更好。虽然Erdős不同距离问题的解决将意味着我们的非固定猜想成立，但Erdős或Falconer距离问题的任何结果似乎都不蕴含我们的固定变体的任何结果。我们将该问题视为一个模型问题，可能最终会对原始的固定距离问题做出贡献。
英文摘要
In this paper we propose and study what we call the Erdős-Falconer distance problem in the integer lattice. One motivation is Magyar's discrete analog of the result of Furstenberg, Katznelson, and Weiss, who showed that all arbitrarily large distances had to appear for a set of positive upper Banach density in $\mathbb{R}^d$, $d\geq 2$. Another motivation is the Falconer distance problem, which can be viewed as a refinement of a result of Steinhaus, dropping assumptions from positive Lebesgue measure to merely asking about the dimension of sets. In this context we thus want to lower the assumption of positive upper Banach density and yet obtain statements that guarantee that we have lots of distinct distances. All of these questions are also related to the Erdős distinct distance problem. Our naming is as for the finite field variant of the problem, acknowledging that parts of the problem share similarities with the Erdős distinct distance problem while others share similarities with the Falconer distance problem. The main tool we use are $\ell^p$-improving inequalities for discrete spherical averages, which to the best of our knowledge is the first application of such bounds. One can use results on the Erdős distinct distance problem to make progress on our conjecture but surprisingly, even if using the strongest available bounds in high dimensions using the mechanism of Solymosi and Vu, then in certain regimes our methods do better. While a resolution of the Erdős distinct distance problem would imply our unpinned conjecture then no results on either the Erdős or Falconer distance problem seem to imply any results for our pinned variant. We view this problem as a model problem that might ultimately contribute back to the original pinned distance problems.
Comments25 pages, 5 figures
