Freiman’s 2.4-Theorem states that any set satisfying and can be covered by an arithmetic progression of length at most . A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying as long as the rather strong density requirement is satisfied. We present a version of this statement that allows for sets satisfying with the more modest density requirement of .
Le théorème de Freiman affirme que tout ensemble qui satisfait les conditions et peut être couvert par une suite arithmétique de longueur inférieure ou égale à . Un résultat plus général de Green et Ruzsa implique que cette propriété de couverture est valable pour tout ensemble qui satisfait et la condition de densité très forte . Nous présentons une version de ce résultat pour tous les ensembles qui satisfont avec la condition de densité plus faible .
Revised:
Accepted:
Published online:
Classification: 11P70, 11B13, 05B10
Keywords: Additive Combinatorics, Sumset, Small Doubling, Inverse Result
Author's affiliations:
@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/} }
TY - JOUR TI - A step beyond Freiman’s theorem for set addition modulo a prime JO - Journal de Théorie des Nombres de Bordeaux PY - 2020 DA - 2020/// SP - 275 EP - 289 VL - 32 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1122/ UR - https://doi.org/10.5802/jtnb.1122 DO - 10.5802/jtnb.1122 LA - en ID - JTNB_2020__32_1_275_0 ER -
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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