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

Candela, Pablo; Serra, Oriol; Spiegel, Christoph

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

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

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

Classification : 11P70, 11B13, 05B10
Mots clés : Additive Combinatorics, Sumset, Small Doubling, Inverse Result

[1] Candela, Pablo; De Roton, A. On sets with small sumset in the circle, Q. J. Math, Volume 70 (2018) no. 1, pp. 49-69 | DOI | MR | Zbl

[2] Candela, Pablo; González-Sánchez, Diego; Grynkiewicz, David J. On sets with small sumset and m-sumfree sets in $\mathbb{Z}/p\mathbb{Z}$ (2019) (https://arxiv.org/abs/1909.07967)

[3] Chang, Mei-Chu A polynomial bound in Freiman’s theorem, Duke Math. J., Volume 113 (2002) no. 3, pp. 399-419 | DOI | MR | Zbl

[4] Deshouillers, Jean-Marc; Freiman, Gregory A. A step beyond Kneser’s theorem for abelian finite groups, Proc. Lond. Math. Soc., Volume 86 (2003) no. 1, pp. 1-28 | DOI | MR | Zbl

[5] Freiman, Gregory A. Inverse problems in additive number theory. Addition of sets of residues modulo a prime, Dokl. Akad. Nauk SSSR, Volume 141 (1961), pp. 571-573 | MR

[6] Freiman, Gregory A. Foundations of a structural theory of set addition, Translations of Mathematical Monographs, 37, American Mathematical Society, 1973 (translated from Russian) | MR | Zbl

[7] Freiman, Gregory A.; Serra, Oriol On doubling and volume: chains, Acta Arith., Volume 186 (2018) no. 1, pp. 37-59 | DOI | MR | Zbl

[8] Green, Ben; Ruzsa, Imre Z. Sets with small sumset and rectification, Bull. Lond. Math. Soc., Volume 38 (2006) no. 1, pp. 43-52 | DOI | MR | Zbl

[9] Green, Ben; Ruzsa, Imre Z. Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc., Volume 75 (2007) no. 1, pp. 163-175 | MR | Zbl

[10] Grynkiewicz, David J. Structural additive theory, Developments in Mathematics, 30, Springer, 2013 | MR | Zbl

[11] Hamidoune, Yahya Ould; Serra, Oriol; Zémor, Gilles On the critical pair theory in $\mathbb{Z}/p\mathbb{Z}$, Acta Arith., Volume 121 (2006) no. 2, pp. 99-115 | DOI | MR | Zbl

[12] Jin, Renling Freiman’s inverse problem with small doubling property, Adv. Math., Volume 216 (2007) no. 2, pp. 711-752 | MR | Zbl

[13] Kemperman, Johannes H. B. On small sumsets in an abelian group, Acta Math., Volume 103 (1960) no. 1-2, pp. 63-88 | DOI | MR | Zbl

[14] Lev, Vsevolod F. Distribution of points on arcs, Integers, Volume 5 (2005) no. 2, A11, 6 pages | MR | Zbl

[15] Lev, Vsevolod F.; Smeliansky, Pavel Y. On addition of two distinct sets of integers, Acta Arith., Volume 70 (1995) no. 1, pp. 85-91 | MR | Zbl

[16] Rødseth, Øystein J. On Freiman’s 2.4-theorem, Skr., K. Nor. Vidensk. Selsk., Volume 2006 (2006) no. 4, pp. 11-18 | MR | Zbl

[17] Ruzsa, Imre Z. Generalized arithmetical progressions and sumsets, Acta Math. Hung., Volume 65 (1994) no. 4, pp. 379-388 | DOI | MR | Zbl

[18] Sanders, Tom Appendix to “Roth’s theorem on progressions revisited” by J. Bourgain, J. Anal. Math., Volume 104 (2008), pp. 193-206 | DOI | MR | Zbl

[19] Schoen, Tomasz Near optimal bounds in Freiman’s theorem, Duke Math. J., Volume 158 (2011) no. 1, pp. 1-12 | DOI | MR | Zbl

[20] Serra, Oriol; Zémor, Gilles Large sets with small doubling modulo $p$ are well covered by an arithmetic progression, Ann. Inst. Fourier, Volume 59 (2009) no. 5, pp. 2043-2060 | DOI | MR | Zbl

[21] Vosper, A. G. The critical pairs of subsets of a group of prime order, J. Lond. Math. Soc., Volume 31 (1956), pp. 200-205 | DOI | MR | Zbl