Abstract
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.
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The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively.
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Lyall, N., Magyar, Á. Distances and trees in dense subsets of ℤd. Isr. J. Math. 240, 769–790 (2020). https://doi.org/10.1007/s11856-020-2079-8
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DOI: https://doi.org/10.1007/s11856-020-2079-8