Ensembles de petite somme, structure de sous-criticité
Robin Riblet1
1 26 rue du Caire 75002 Paris, France
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 697-749.

En 1991, Ruzsa a démontré une minoration précise de la mesure de la somme de deux ensembles bornés de réels A et B faisant intervenir le ratio λ(A)/λ(B) et le diamètre de B. La structure des ensembles critiques pour cette minoration a été décrite par Roton en 2018. Dans cet article, nous décrivons la structure des ensembles presque critiques pour l’inégalité de Ruzsa.

In 1991, Ruzsa proved a precise lower bound for the measure of the sum of two bounded sets of real numbers A and B involving the ratio λ(A)/λ(B) and the diameter of B. The structure of critical sets for this lower bound was described by Roton in 2018. In this paper, we describe the structure of nearly critical sets for Ruzsa’s inequality.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1261
Classification : 28A75, 11B13, 05B10
Mots clés : Combinatoire additive, somme de Minkowski, structure, mesure d’ensembles, problème inverse, ensembles critiques, sumsets
Robin Riblet 1

1 26 rue du Caire 75002 Paris, France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2023__35_3_697_0,
     author = {Robin Riblet},
     title = {Ensembles de petite somme, structure de sous-criticit\'e},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {697--749},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {35},
     number = {3},
     year = {2023},
     doi = {10.5802/jtnb.1261},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1261/}
}
TY  - JOUR
AU  - Robin Riblet
TI  - Ensembles de petite somme, structure de sous-criticité
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2023
SP  - 697
EP  - 749
VL  - 35
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1261/
DO  - 10.5802/jtnb.1261
LA  - fr
ID  - JTNB_2023__35_3_697_0
ER  - 
%0 Journal Article
%A Robin Riblet
%T Ensembles de petite somme, structure de sous-criticité
%J Journal de théorie des nombres de Bordeaux
%D 2023
%P 697-749
%V 35
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1261/
%R 10.5802/jtnb.1261
%G fr
%F JTNB_2023__35_3_697_0
Robin Riblet. Ensembles de petite somme, structure de sous-criticité. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 697-749. doi : 10.5802/jtnb.1261. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1261/

[1] Yuri Bilu The (α+2β)-inequality on a torus, J. Lond. Math. Soc., Volume 57 (1998) no. 3, pp. 513-528 | DOI | MR | Zbl

[2] Pablo Candela; Anne de Roton On sets with small sumset in the circle, Q. J. Math., Volume 70 (2019) no. 1, pp. 49-69 | DOI | MR | Zbl

[3] Pablo Candela; Oriol Serra; Christoph Spiegel A step beyond Freiman’s theorem for set addition modulo a prime, J. Théor. Nombres Bordeaux, Volume 32 (2020) no. 1, pp. 275-289 | DOI | Numdam | MR | Zbl

[4] Augustin L. Cauchy Recherche sur les nombres, J. Éc. Polytech., Math., Volume 9 (1813), pp. 99-116

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

[6] Harold Davenport On the addition of residue classes, J. Lond. Math. Soc., Volume 10 (1935), pp. 30-32 | DOI | MR | Zbl

[7] Harold Davenport A historical note, J. Lond. Math. Soc., Volume 22 (1947), pp. 100-101 | DOI | MR | Zbl

[8] Gregory A. Freiman The addition of finite sets, Izv. Vyssh. Uchebn. Zaved., Mat., Volume 1964 (1959) no. 6, pp. 202-213 | MR | Zbl

[9] 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

[10] Gregory A. Freiman Foundations of a Structural Theory of Set Addition, Kazan Gos. Ped. Inst., 1966, 140 pages

[11] 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

[12] 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

[13] David J. Grynkiewicz Structural additive theory, Developments in Mathematics, 30, Springer, 2013 | DOI

[14] J. H. B. Kemperman On products of sets in a locally compact group, Fundam. Math., Volume 56 (1964), pp. 51-68 | DOI | MR | Zbl

[15] Martin Kneser Summenmengen in lokalkompakten abelschen Gruppen, Math. Z., Volume 66 (1956), pp. 88-110 | DOI | MR | Zbl

[16] Melvyn B. Nathanson Additive Number Theory : Inverse Problems and Geometry of Sumsets, Graduate Texts in Mathematics, 165, Springer, 1996

[17] Robin Riblet Ensembles de petite somme et ensembles de Sidon, étude de deux extrêmes, Ph. D. Thesis, Université de Lorraine, École Polytechnique (2021)

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

[19] Anne de Roton Small sumsets in ℝ : full continuous 3k-4 theorem, critical sets, J. Éc. Polytech., Math., Volume 5 (2018), pp. 177-196 | DOI | MR | Zbl

[20] Imre Z. Ruzsa Diameter of sets and measure of sumsets, Monatsh. Math., Volume 112 (1991) no. 4, pp. 323-328 | DOI | MR | Zbl

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

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

[23] Tom Sanders On the Bogolyubov–Ruzsa lemma, Anal. PDE, Volume 5 (2010) no. 3, pp. 627-655 | DOI | MR | Zbl

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

[25] 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 | Numdam | MR | Zbl

[26] Terence Tao An inverse Theorem for an inequality of Kneser, Proc. Steklov Inst. Math., Volume 303 (2018), pp. 193-219 | MR | Zbl

[27] Terence Tao; Van H. Vu Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, 2006 | DOI

[28] A. J. Vosper Addendum to "The critical pairs of subsets of a group of prime order", J. Lond. Math. Soc., Volume 31 (1956), pp. 280-282 | DOI | MR

[29] A. J. 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

Cité par Sources :