Ensembles de petite somme, structure de sous-criticité

Riblet, Robin

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 $\lambda \left(A\right)/\lambda \left(B\right)$ 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 $\lambda \left(A\right)/\lambda \left(B\right)$ 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.

Classification : 28A75, 11B13, 05B10
Mots-clés : Combinatoire additive, somme de Minkowski, structure, mesure d’ensembles, problème inverse, ensembles critiques, sumsets

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