Improving Roth's theorem in the primes

2009/12/09 by Harald Andres Helfgott, Helfgott, Harald Andres, Anne de Roton +1
Mathematics · #11B25 #11P32 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B25 #msc:11P32

paper · pdf · doi:10.48550/arxiv.0912.1842

13 pages

arxiv created 2009/12/09 · arxiv updated 2010/01/07

Abstract

Let A be a subset of the primes. Let δ_P(N) = \frac|\n∈ A: n≤ N\||\n prime: n≤ N\|. We prove that, if δ_P(N)≥ C \fraclog log log N(log log N)^1/3 for N≥ N₀, where C and N₀ are absolute constants, then A∩ [1,N] contains a non-trivial three-term arithmetic progression. This improves on B. Green's result, which needs δ_P(N) ≥ C' √((log log log log log N)/(log log log log N)).

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• A Density Increment Approach to Roth's Theorem in the Primes
• Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions
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• Roth's Theorem in the Piatetski-Shapiro primes