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 ; Oriol Serra ; Christoph Spiegel

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

Résumé
Abstract

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 $| A | < p / 10^{10}$ .

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 $| A | < p / 10^{10}$ .

Reçu le : 2019-06-28
Révisé le : 2019-10-07
Accepté le : 2019-10-25
Publié le : 2020-08-21

DOI : https://doi.org/10.5802/jtnb.1122
Classification : 11P70, 11B13, 05B10
Mots clés : Additive Combinatorics, Sumset, Small Doubling, Inverse Result

@article{JTNB_2020__32_1_275_0,
     author = {Pablo Candela and Oriol Serra and Christoph Spiegel},
     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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1122},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1122/}
}

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, Tome 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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