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 $(A+A)\cap [2c_n,2d_n]=W\cap [2c_n,2d_n]$.

[1] Y. Bilu, Addition of sets of integers of positive density. The Journal of Number Theory 64 (1997), No. 2, 233–275. | MR | Zbl

[2] Y. Bilu, Structure of sets with small sumset. Asterisque 258 (1999), 77–108. | Numdam | MR | Zbl

[3] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, 1981. | MR | Zbl

[4] H. Halberstam, K. F. Roth, Sequences. Oxford University Press, 1966. | MR | Zbl

[5] C. W. Henson, Foundations of nonstandard analysis–A gentle introduction to nonstandard extension in Nonstandard Analysis: Theory and Applications. Ed. by N. J. Cutland, C. W. Henson, and L. Arkeryd. Kluwer Academic Publishers, 1997. | MR | Zbl

[6] R. Jin, Nonstandard methods for upper Banach density problems. The Journal of Number Theory, 91 (2001), 20–38. | MR | Zbl

[7] R. Jin, Standardizing nonstandard methods for upper Banach density problems in the DIMACS series Unusual Applications of Number Theory, edited by M. Nathanson. Vol. 64 (2004), 109–124. | MR | Zbl

[8] R. Jin, Inverse problem for upper asymptotic density. The Transactions of American Mathematical Society 355 (2003), No. 1, 57–78. | MR | Zbl

[9] R. Jin, Solution to the Inverse problem for upper asymptotic density. Journal für die reine und angewandte Mathematik 595 (2006), 121–166. | MR | Zbl

[10] T. Lindstrom, An invitation to nonstandard analysis in Nonstandard Analysis and Its Application. Ed. by N. Cutland. Cambridge University Press, 1988. | MR | Zbl

[11] M. B. Nathanson, Additive Number Theory–Inverse Problems and the Geometry of Sumsets. Springer, 1996. | MR | Zbl

[12] K. Petersen, Ergodic Theory. Cambridge University Press, 1983. | MR | Zbl