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This work has benefited from support from the Spanish Ministerio de Ciencia e Innovacion project MTM2017-83496-P and from the La Caixa Foundation (ID 100010434) under agreement LCF/BQ/SO16/52270027.

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

Publicado en:BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE. 149 (1): 155-177 - 2021-01-01 149(1), DOI: 10.24033/bsmf.2827

Autores: Candela, Pablo; Gonzalez-Sanchez, Diego; Grynkiewicz, David J.;

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The 3k - 4 conjecture in groups Z/pZ for p prime states that if A is a nonempty subset of Z/pZ satisfying 2A not equal Z/pZ and vertical bar 2A vertical bar = 2 vertical bar A vertical bar + r <= min{3 vertical bar A vertical bar - 4, p - r - 4}, then A is covered by an arithmetic progression of size at most vertical bar A vertical bar + r + 1. Previously, the best result toward this conjecture, without any additional constraint on vertical bar A vertical bar, was a theorem of Serra and Zemor proving the conjecture provided r <= 0.0001 vertical bar A vertical bar. Subject to the mild additional constraint vertical bar 2A vertical bar <= 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 vertical bar A vertical bar. 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) is an element of A(3) 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-Zemor 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 ->infinity) Another application of our main results concerns sets of the form A+A/A in F-p, and we also improve the structural description of large sum-free sets in Z/pZ.

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