Arithmetic progressions in Salem-type subsets of the integers

2010/05/07 by Paul Potgieter, Potgieter, Paul
Mathematics · #11B25 #26E35 #28A78 #42B05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B25 #msc:26E35 #msc:28A78 #msc:42B05

paper · pdf · doi:10.48550/arxiv.1005.1210

14 pages

arxiv created 2010/07/13 · arxiv updated 2010/07/14

Abstract

Given a subset of the integers of zero density, we define the weaker notion of fractional density of such a set. It is shown how this notion corresponds to that of the Hausdorff dimension of a compact subset of the reals. We then show that a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density and satisfying certain Fourier-decay conditions.

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