Kneser’s theorem for upper Banach density
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 α (voir définition plus bas) et que la densité de Banach supérieure de A+A soit inférieure à 2α. 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 [2αg-1] suites arithmétiques [We call a set of the form a+d 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+AW et si [a n ,b n ] est, pour chaque n, un intervalle d’entiers tel que b n -a n et la densité relative de A dans [a n ,b n ] approche α, il existe alors un intervalle [c n ,d n ][a n ,b n ] pour chaque n tel que (d n -c n )/(b n -a n )1 et (A+A)[2c n ,2d n ]=W[2c n ,2d n ].

Suppose A is a set of non-negative integers with upper Banach density α (see definition below) and the upper Banach density of A+A is less than 2α. 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 2αg-1 arithmetic sequences [We call a set of the form a+d 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+AW and if [a n ,b n ] for each n is an interval of integers such that b n -a n and the relative density of A in [a n ,b n ] approaches α, then there is an interval [c n ,d n ][a n ,b n ] for each n such that (d n -c n )/(b n -a n )1 and (A+A)[2c n ,2d n ]=W[2c n ,2d n ].

DOI : https://doi.org/10.5802/jtnb.547
Classification : 11B05,  11B13,  11U10,  03H15
Mots clés : Upper Banach density, inverse problem, nonstandard analysis
@article{JTNB_2006__18_2_323_0,
     author = {Bihani, Prerna and Jin, Renling},
     title = {Kneser{\textquoteright}s theorem for upper Banach density},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {323--343},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {2},
     year = {2006},
     doi = {10.5802/jtnb.547},
     mrnumber = {2289427},
     zbl = {05135393},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.547/}
}
Bihani, Prerna; Jin, Renling. Kneser’s theorem for upper Banach density. Journal de Théorie des Nombres de Bordeaux, Tome 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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