A step beyond Freiman’s theorem for set addition modulo a prime

Pablo Candela; Oriol Serra; Christoph Spiegel

doi:10.5802/jtnb.1122

Pablo Candela ¹; Oriol Serra ²; Christoph Spiegel ³

¹ Universidad Autónoma de Madrid and ICMAT Ciudad Universitaria de Cantoblanco, Madrid 28049, Spain
² Universitat Politécnica de Catalunya Department of Mathematics, 08034 Barcelona, Spain
³ Universitat Politécnica de Catalunya and Barcelona Graduate School of Mathematics Department of Mathematics, Edificio Omega, 08034 Barcelona, Spain

Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 275-289.

Abstract
Résumé

Freiman’s 2.4-Theorem states that any set $A \subset ℤ_{p}$ satisfying $| 2 A | \leq 2.4 | A | - 3$ and $| A | < p / 35$ can be covered by an arithmetic progression of length at most $| 2 A | - | A | + 1$ . A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying $| 2 A | \leq 3 | A | - 4$ as long as the rather strong density requirement $| A | < p / 10^{215}$ is satisfied. We present a version of this statement that allows for sets satisfying $| 2 A | \leq 2.48 | A | - 7$ with the more modest density requirement of $| A | < p / 10^{10}$ .

Le théorème $2.4$ de Freiman affirme que tout ensemble $A \subset ℤ_{p}$ qui satisfait les conditions $| 2 A | \leq 2.4 | A | - 3$ et $| A | < p / 35$ peut être couvert par une suite arithmétique de longueur inférieure ou égale à $| 2 A | - | A | + 1$ . Un résultat plus général de Green et Ruzsa implique que cette propriété de couverture est valable pour tout ensemble qui satisfait $| 2 A | \leq 3 | A | - 4$ et la condition de densité très forte $| A | < p / 10^{215}$ . Nous présentons une version de ce résultat pour tous les ensembles qui satisfont $| 2 A | \leq 2.48 | A | - 7$ avec la condition de densité plus faible $| A | < p / 10^{10}$ .

Received: 2019-06-28
Revised: 2019-10-07
Accepted: 2019-10-25
Published online: 2020-08-21

DOI: 10.5802/jtnb.1122
Classification: 11P70, 11B13, 05B10
Keywords: Additive Combinatorics, Sumset, Small Doubling, Inverse Result
Author's affiliations:

Pablo Candela ¹; Oriol Serra ²; Christoph Spiegel ³

¹ Universidad Autónoma de Madrid and ICMAT Ciudad Universitaria de Cantoblanco, Madrid 28049, Spain
² Universitat Politécnica de Catalunya Department of Mathematics, 08034 Barcelona, Spain
³ Universitat Politécnica de Catalunya and Barcelona Graduate School of Mathematics Department of Mathematics, Edificio Omega, 08034 Barcelona, Spain

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     title = {A step beyond {Freiman{\textquoteright}s} theorem for set addition modulo a prime},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {275--289},
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Pablo Candela; Oriol Serra; Christoph Spiegel. A step beyond Freiman’s theorem for set addition modulo a prime. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 275-289. doi : 10.5802/jtnb.1122. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1122/

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