A survey on additive and multiplicative decompositions of sumsets and of shifted sets

Abstract

In this paper we survey results on sumsets with multiplicative properties and the question if a shifted copy of a multiplicatively defined set can again be multiplicatively defined. The methods involved are of analytic nature such as the large sieve, and of combinatorial nature such as extremal graph theory.

Counting additive decompositions of quadratic residues in finite fields

“…Furthermore, Dartyge and Sárközy [1] have made a similar conjecture for the set R of primitive roots modulo p. We also refer to [1,2,6] for further references about set decompositions.…”

On an inverse ternary Goldbach problem

“…Indeed, if this is true for any δ < 1/2 then the inverse (binary) Goldbach problem follows. In the other direction, there exist A 1 , A 2 ⊂ [N] with |A 1 |, |A 2 | ≥ log N/ log log N such that A 1 + A 2 is contained in the primes (see Corollary 1.3.6 in [6]). Note also the similarity between Conjecture 1.2 and the problem of finding the clique numbers of Paley sum graphs, constructed using quadratic residues in a finite field.…”

“…Its ternary analogue is solved by Elsholtz [4]. For more references on this problem see [5,7] and the survey [6]. In this paper we study additive decompositions of subsets of primes.…”