On Szemerédi's theorem with differences from a random set

2019/05/13 by Altman, Daniel · 1 citation
#11B25 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1905.05045

Abstract

We consider, over both the integers and finite fields, Szemerédi's theorem on k-term arithmetic progressions where the set S of allowed common differences in those progressions is restricted and random. Fleshing out a line of enquiry suggested by Frantzikinakis et al, we show that over the integers, the conjectured threshold for ℙ(d ∈ S) for Szemerédi's theorem to hold a.a.s follows from a conjecture about how so-called dual functions are approximated by nilsequences. We also show that the threshold over finite fields is different to this threshold over the integers.

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• A Note on Lower Bounds in Szemerédi's Theorem with Random Differences

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