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Improving Roth's theorem in the primes

Research paper by Harald Andres Helfgott, Anne de Roton

Indexed on: 09 Dec '09Published on: 09 Dec '09Published in: Mathematics - Number Theory



Abstract

Let A be a subset of the primes. Let \delta_P(N) = \frac{|\{n\in A: n\leq N\}|}{|\{\text{$n$ prime}: n\leq N\}|}. We prove that, if \delta_P(N)\geq C \frac{\log \log \log N}{(\log \log N)^{1/3}} for N\geq N_0, where C and N_0 are absolute constants, then A\cap [1,N] contains a non-trivial three-term arithmetic progression. This improves on B. Green's result, which needs \delta_P(N) \geq C' \sqrt{\frac{\log \log \log \log \log N}{\log \log \log \log N}}.