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On an inverse ternary Goldbach problem (1404.6022v2)

Published 24 Apr 2014 in math.NT

Abstract: We prove an inverse ternary Goldbach-type result. Let $N$ be sufficiently large and $c>0$ be sufficiently small. If $A_1,A_2,A_3\subset [N]$ are subsets with $|A_1|,|A_2|,|A_3|\geq N{1/3-c}$, then $A_1+A_2+A_3$ contains a composite number. This improves on the bound $N{1/3}$ from Gallagher's larger sieve. The main ingredients in our argument include a type of inverse sieve result in the larger sieve regime, and a variant of the analytic large sieve inequality.

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