In [4] Furstenberg, Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of ℝ2 of positive upper (Banach) density. A second proof of this result, as well as a stronger “pinned variant”, was given by Bourgain in [2] using Fourier analytic methods. In [8] the second author adapted Bourgain’s Fourier analytic approach to establish a result analogous to that of Furstenberg, Katznelson and Weiss for subsets ℤd provided d ≥ 5. We present a new direct proof of this discrete distance set result and generalize this to arbitrary trees. Using appropriate discrete spherical maximal function theorems we ultimately establish the natural “pinned variants” of these results.

中文翻译：

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ℤd 的密集子集中的距离和树

在 [4] Furstenberg 中，Katznelson 和 Weiss 建立了所有足够大的距离总是可以在来自正上（Banach）密度的 ℝ2 的任何给定可测量子集的点对之间获得。Bourgain 在 [2] 中使用傅立叶分析方法给出了该结果的第二个证明以及更强的“固定变体”。在 [8] 中，第二作者采用 Bourgain 的傅立叶分析方法来建立类似于 Furstenberg、Katznelson 和 Weiss 的结果，子集 ℤd 提供 d ≥ 5。我们提出了这个离散距离集结果的新直接证明，并将其推广到任意树木。使用适当的离散球面极大函数定理，我们最终建立了这些结果的自然“固定变体”。