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

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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
AU  - Pablo Candela
AU  - Oriol Serra
AU  - Christoph Spiegel
TI  - A step beyond Freiman’s theorem for set addition modulo a prime
JO  - Journal de théorie des nombres de Bordeaux
PY  - 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/
DO  - 10.5802/jtnb.1122
LA  - en
ID  - JTNB_2020__32_1_275_0
ER  -

%0 Journal Article
%A Pablo Candela
%A Oriol Serra
%A Christoph Spiegel
%T A step beyond Freiman’s theorem for set addition modulo a prime
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 275-289
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1122/
%R 10.5802/jtnb.1122
%G en
%F JTNB_2020__32_1_275_0

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/

References
Cited by

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[20] Oriol Serra; Gilles Zémor 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 | Numdam | Zbl

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

Cited by Sources: