A step towards the 3k - 4 conjecture in Z/pZ and an application to m-sum-free sets

The 3k - 4 conjecture in Z/pZ states that if A is a nonempty subset of Z/pZ satisfying 2A ≠Z/pZ and |2A| = 2|A| + r ≤ min{3|A| - 4, p - r - 3}, then A is covered by an arithmetic progression of size at most |A | + r + 1. In this paper we summarize progress made towards this conjecture in a recent joint paper of the same authors. In that paper we prove first that if |2A| ≤ (2 + α)|A| — 3 for α ≈ 0.136861 and |2A| ≤ 3p/4, then A is efficiently covered by an arithmetic progression, as in the conclusion of the conjecture. With a refined argument we prove that we can go up to α = (image, found) (1) at the cost of restricting |A| ≤ (p - r)/3. We then use this to investigate the maximum size of m-sum-free sets for m ≥ 3, i.e., sets A m Z/p Z such that the equation x + y = mz has no solution in A. We obtain that for m fixed, limp→∞ max{|A|/p: A C Z/pZ m-sum-free} ≤ 1/3.1955 (previously, the best known upper bound was 1/3.0001)