Distances and Trees in Dense Subsets of ℤ^d

2015/09/30 by Lyall, Neil, Magyar, Akos
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1509.09298

Abstract

In \citeFKW 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 \citeB using Fourier analytic methods. In \citeM1 the second author adapted Bourgain's Fourier analytic approach to established a result analogous to that of Katznelson and Weiss for subsets ℤ^d 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.

Related

• The Erdos-Falconer distance problem in the tree setting
• Spherical Recurrence and locally isometric embeddings of trees into positive density subsets of ℤ^d
• Nearly equal distances in the plane, II
• On distance measures for well-distributed sets
• On Ramsey-type properties of the distance in nonseparable spheres