Kneser’s theorem for upper Banach density

Bihani, Prerna; Jin, Renling

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 323-343.

Supposons que $A$ soit un ensemble d’entiers non négatifs avec densité de Banach supérieure $\alpha$ (voir définition plus bas) et que la densité de Banach supérieure de $A+A$ soit inférieure à $2\alpha$. Nous caractérisons la structure de $A+A$ en démontrant la proposition suivante : il existe un entier positif $g$ et un ensemble $W$ qui est l’union des $[2\alpha g-1]$ suites arithmétiques [We call a set of the form $a+d\mathbb{N}$ an arithmetic sequence of difference $d$ and call a set of the form $\{a,a+d,a+2d,...,a+kd\}$ an arithmetic progression of difference $d$. So an arithmetic progression is finite and an arithmetic sequence is infinite.] avec la même différence $g$ tels que $A+A\subseteq W$ et si $[a_n,b_n]$ est, pour chaque $n$, un intervalle d’entiers tel que $b_n-a_n\to \infty$ et la densité relative de $A$ dans $[a_n,b_n]$ approche $\alpha$, il existe alors un intervalle $[c_n,d_n]\subseteq [a_n,b_n]$ pour chaque $n$ tel que $(d_n-c_n)/(b_n-a_n)\to 1$ et $(A+A)\cap [2c_n,2d_n]=W\cap [2c_n,2d_n]$.

Suppose $A$ is a set of non-negative integers with upper Banach density $\alpha$ (see definition below) and the upper Banach density of $A+A$ is less than $2\alpha$. We characterize the structure of $A+A$ by showing the following: There is a positive integer $g$ and a set $W$, which is the union of $\lceil 2\alpha g-1\rceil$ arithmetic sequences [We call a set of the form $a+d\mathbb{N}$ an arithmetic sequence of difference $d$ and call a set of the form $\{a,a+d,a+2d,...,a+kd\}$ an arithmetic progression of difference $d$. So an arithmetic progression is finite and an arithmetic sequence is infinite.] with the same difference $g$ such that $A+A\subseteq W$ and if $[a_n,b_n]$ for each $n$ is an interval of integers such that $b_n-a_n\to \infty$ and the relative density of $A$ in $[a_n,b_n]$ approaches $\alpha$, then there is an interval $[c_n,d_n]\subseteq [a_n,b_n]$ for each $n$ such that $(d_n-c_n)/(b_n-a_n)\to 1$ and $(A+A)\cap [2c_n,2d_n]=W\cap [2c_n,2d_n]$.

Classification : 11B05, 11B13, 11U10, 03H15
Mots clés : Upper Banach density, inverse problem, nonstandard analysis

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