On sets with small sumset and m-sum-free sets in Z/pZ

The 3k - 4 conjecture in groups Z/pZ for p prime states that if A is a nonempty subset of Z/pZ satisfying 2A 6= Z/pZ and |2A| = 2|A| + r ≤ min{3|A| - 4, p - r - 4}, then A is covered by an arithmetic progression of size at most |A| + r + 1. Previously, the best result toward this conjecture, without any additional constraint on |A|, was a theorem of Serra and Zémor proving the conjecture provided r ≤ 0.0001|A|. Subject to the mild additional constraint |2A| ≤ 3p/4, which is optimal in the sense explained in the paper, our first main result improves the bound on r, allowing r ≤ 0.1368|A|. We also prove a variant that further improves this bound on r provided that A is sufficiently dense. We then give several applications. First, we apply the above variant to give a new upper bound for the maximal density of m-sum-free sets in Z/pZ, i.e., sets A having no solution (x, y, z) ∈ A3 to the equation x + y = mz, where m ≥ 3 is a fixed integer. The previous best upper bound for this maximal density was 1/3.0001 (using the Serra-Zémor theorem). We improve this to 1/3.1955. We also present a construction following an idea of Schoen, which yields a lower bound for this maximal density of the form 1/8 + o(1)p→∞. Another application of our main results concerns sets of the form A+AA in Fp, and we also improve the structural description of large sum-free sets in Z/pZ