A new bound for A(A + A) for large sets

2020/11/23 by Semchankau, Aliaksei
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2011.11468

Abstract

For p being a large prime number, and A ⊂ \mathbbF_p we prove the following: (i) If A(A+A) does not cover all nonzero residues in \mathbbF_p, then |A| < p/8 + o(p). (ii) If A is both sum-free and satisfies A = A^*, then |A| < p/9 + o(p). (iii) If |A| ≫ \fracloglogp√logpp, then |A + A^*| \geqslant (1 - o(1))min(2√(|A|p), p). Here the constants 1/8, 1/9, and 2 are the best possible. The proof involves wrappers, subsets of a finite abelian group G, with which we `wrap' popular values in convolutions A * B for dense sets A, B ⊆ G. These objects carry some special structural features, making them capable of addressing both additive-combinatorial and enumerative problems.

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