Improved bounds for five-term arithmetic progressions

2023/12/17 by Leng, James, Sah, Ashwin, Sawhney, Mehtaab · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.10776

Abstract

Let r₅(N) be the largest cardinality of a set in \1,…,N\ which does not contain 5 elements in arithmetic progression. Then there exists a constant c∈ (0,1) such that r₅(N)≪ \fracNexp((loglog N)^c). Our work is a consequence of recent improved bounds on the U⁴-inverse theorem of the first author and the fact that 3-step nilsequences may be approximated by locally cubic functions on shifted Bohr sets. This combined with the density increment strategy of Heath-Brown and Szemerédi, codified by Green and Tao, gives the desired result.

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