[Ensembles de réels de petite somme : une version continue du théorème 3k-4, structure des ensembles critiques]
Nous démontrons un théorème , 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 , et lorsque et soit la mesure de est supérieure à celle de , soit le diamètre de est supérieur à celui de . 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.
We prove a full continuous Freiman’s 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 , and when and either or has larger diameter than . 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 .
Accepté le :
Publié le :
DOI : 10.5802/jep.67
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
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@article{JEP_2018__5__177_0,
author = {Anne de Roton},
title = {Small sumsets in $\protect \mathbb{R}$: full continuous $3k-4$ theorem, critical sets},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {177--196},
publisher = {\'Ecole polytechnique},
volume = {5},
year = {2018},
doi = {10.5802/jep.67},
zbl = {06988577},
mrnumber = {3738512},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.67/}
}
TY - JOUR
AU - Anne de Roton
TI - Small sumsets in $\protect \mathbb{R}$: full continuous $3k-4$ theorem, critical sets
JO - Journal de l’École polytechnique — Mathématiques
PY - 2018
SP - 177
EP - 196
VL - 5
PB - École polytechnique
UR - https://jep.centre-mersenne.org/articles/10.5802/jep.67/
DO - 10.5802/jep.67
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%D 2018
%P 177-196
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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, Tome 5 (2018), pp. 177-196. doi : 10.5802/jep.67. https://jep.centre-mersenne.org/articles/10.5802/jep.67/
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