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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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Candela, PCorresponding AuthorGonzalez-Sanchez, DAuthor

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May 24, 2021
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On sets with small sumset and m-sum-free sets in Z/pZ

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

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

Affiliations

‎ ICMAT, Madrid 28049, Spain - Author
‎ Univ Autonoma Madrid, Madrid 28049, Spain - Author
‎ Univ Memphis, Dept Math Sci, Memphis, TN 38152 USA - Author

Abstract

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.

Keywords

3k3k-4 conjecture4 conjecture4 theoremAdditive combinatoricsFreiman's 3k-4 theoremFreiman’s 3kM-sum-free setModuloSmall sumsetTheorem

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE due to its progression and the good impact it has achieved in recent years, according to the agency Scopus (SJR), it has become a reference in its field. In the year of publication of the work, 2021, it was in position , thus managing to position itself as a Q2 (Segundo Cuartil), in the category Mathematics (Miscellaneous). Notably, the journal is positioned en el Cuartil Q4 for the agency WoS (JCR) in the category Mathematics.

Independientemente del impacto esperado determinado por el canal de difusión, es importante destacar el impacto real observado de la propia aportación.

Según las diferentes agencias de indexación, el número de citas acumuladas por esta publicación hasta la fecha 2025-08-03:

  • WoS: 1
  • Scopus: 1

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-08-03:

With a more dissemination-oriented intent and targeting more general audiences, we can observe other more global scores such as:

    It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

    • The work has been submitted to a journal whose editorial policy allows open Open Access publication.

    Leadership analysis of institutional authors

    This work has been carried out with international collaboration, specifically with researchers from: United States of America.

    There is a significant leadership presence as some of the institution’s authors appear as the first or last signer, detailed as follows: First Author (Candela Pokorna, Pablo) .

    the author responsible for correspondence tasks has been Candela Pokorna, Pablo.