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On an inverse ternary Goldbach problem
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 5, October 2016
- pp. 1167-1191
- 10.1353/ajm.2016.0038
- Article
- Additional Information
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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 $N^{1/3+o(1)}$ obtained by using 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.