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.

Freiman’s 2.4-Theorem states that any set A p satisfying |2A|2.4|A|-3 and |A|<p/35 can be covered by an arithmetic progression of length at most |2A|-|A|+1. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying |2A|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 |2A|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 p qui satisfait les conditions |2A|2.4|A|-3 et |A|<p/35 peut être couvert par une suite arithmétique de longueur inférieure ou égale à |2A|-|A|+1. Un résultat plus général de Green et Ruzsa implique que cette propriété de couverture est valable pour tout ensemble qui satisfait |2A|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 |2A|2.48|A|-7 avec la condition de densité plus faible |A|<p/10 10 .

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Published online:
DOI: 10.5802/jtnb.1122
Classification: 11P70,  11B13,  05B10
Keywords: Additive Combinatorics, Sumset, Small Doubling, Inverse Result
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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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