Large subsets of discrete hypersurfaces in Zd contain arbitrarily many collinear points

Joel Moreira, Florian Karl Richter

Research output: Contribution to journalArticlepeer-review

Abstract

In 1977 L.T. Ramsey showed that any sequence in Z2 with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average.We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem. Let d∈N, let f:Zd→Zd+1 be a Lipschitz map and let A⊂Zd have positive upper Banach density. Then f(A) contains arbitrarily many collinear points.Note that Pomerance's theorem corresponds to the special case d= 1. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.

Original languageEnglish (US)
Pages (from-to)163-176
Number of pages14
JournalEuropean Journal of Combinatorics
Volume54
DOIs
StatePublished - May 1 2016

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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