Small sumsets in : full continuous 3k-4 theorem, critical sets
Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 177-196.

We prove a full continuous Freiman’s 3k-4 theorem for small sumsets in by using some ideas from Ruzsa’s work on measure of sumsets in as well as some graphic representation of density functions of sets. We thereby get some structural properties of A, B and A+B when λ(A+B)<λ(A)+2λ(B) and either λ(A)λ(B) or A has larger diameter than B. We also give some structural information for sets of large density according to the size of their sumset, a result so far unknown in the discrete and the continuous setting. Finally, we characterise the critical sets for which equality holds in the lower bounds for λ(A+B).

Nous démontrons un théorème 3k-4, dans sa version la plus complète, pour les ensembles de réels en utilisant des idées issues du travail de Ruzsa sur les mesures des sommes d’ensembles de réels et une représentation graphique liée à la densité des ensembles. Nous obtenons ainsi des informations sur les structures des ensembles A, B et A+B lorsque λ(A+B)<λ(A)+2λ(B) et soit la mesure de A est supérieure à celle de B, soit le diamètre de A est supérieur à celui de B. Nous obtenons aussi des informations sur la structure des ensembles de grande densité en fonction de la taille de leur somme, ce qui représente un résultat n’ayant pas d’analogue discret. Nous caractérisons enfin les ensembles de réels critiques pour lesquels la mesure de l’ensemble somme atteint le minorant que nous avons obtenu.

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Accepted:
Published online:
DOI: 10.5802/jep.67
Classification: 28A75, 11B13, 05B10
Keywords: Sumsets, critical sets, Lebesgue measure, inverse theorems in additive combinatorics
Mot clés : Ensembles sommes, ensembles critiques, mesure de Lebesgue, théorèmes inverses en combinatoire additive
Anne de Roton 1

1 Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, UMR 7502 B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Anne de Roton. Small sumsets in $\protect \mathbb{R}$:  full continuous $3k-4$ theorem, critical sets. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 177-196. doi : 10.5802/jep.67. https://jep.centre-mersenne.org/articles/10.5802/jep.67/

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