Kneser’s theorem for upper Banach density
Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 323-343.

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 $⌈2\alpha g-1⌉$ arithmetic sequences [We call a set of the form $a+dℕ$ an arithmetic sequence of difference $d$ and call a set of the form $\left\{a,a+d,a+2d,...,a+kd\right\}$ 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 $\left[{a}_{n},{b}_{n}\right]$ for each $n$ is an interval of integers such that ${b}_{n}-{a}_{n}\to \infty$ and the relative density of $A$ in $\left[{a}_{n},{b}_{n}\right]$ approaches $\alpha$, then there is an interval $\left[{c}_{n},{d}_{n}\right]\subseteq \left[{a}_{n},{b}_{n}\right]$ for each $n$ such that $\left({d}_{n}-{c}_{n}\right)/\left({b}_{n}-{a}_{n}\right)\to 1$ and $\left(A+A\right)\cap \left[2{c}_{n},2{d}_{n}\right]=W\cap \left[2{c}_{n},2{d}_{n}\right]$.

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 $\left[2\alpha g-1\right]$ suites arithmétiques [We call a set of the form $a+dℕ$ an arithmetic sequence of difference $d$ and call a set of the form $\left\{a,a+d,a+2d,...,a+kd\right\}$ 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 $\left[{a}_{n},{b}_{n}\right]$ est, pour chaque $n$, un intervalle d’entiers tel que ${b}_{n}-{a}_{n}\to \infty$ et la densité relative de $A$ dans $\left[{a}_{n},{b}_{n}\right]$ approche $\alpha$, il existe alors un intervalle $\left[{c}_{n},{d}_{n}\right]\subseteq \left[{a}_{n},{b}_{n}\right]$ pour chaque $n$ tel que $\left({d}_{n}-{c}_{n}\right)/\left({b}_{n}-{a}_{n}\right)\to 1$ et $\left(A+A\right)\cap \left[2{c}_{n},2{d}_{n}\right]=W\cap \left[2{c}_{n},2{d}_{n}\right]$.

DOI: 10.5802/jtnb.547
Classification: 11B05,  11B13,  11U10,  03H15
Keywords: Upper Banach density, inverse problem, nonstandard analysis
Bihani, Prerna 1; Jin, Renling 2

1 Department of Mathematics University of Notre Dame Notre Dame, IN 46556, U.S.A.
2 Department of Mathematics College of Charleston Charleston, SC 29424, U.S.A.
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Bihani, Prerna; Jin, Renling. Kneser’s theorem for upper Banach density. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 323-343. doi : 10.5802/jtnb.547. http://www.numdam.org/articles/10.5802/jtnb.547/

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