Kneser’s theorem for upper Banach density

Prerna Bihani; Renling Jin

doi:10.5802/jtnb.547

Prerna Bihani ¹ ; Renling Jin 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.

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

Résumé
Abstract

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 $2 α$ . 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 $[2 α g - 1]$ suites arithmétiques [We call a set of the form $a + d ℕ$ an arithmetic sequence of difference $d$ and call a set of the form ${a, a + d, a + 2 d, ..., a + k d}$ 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 $α$ , 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 $(A + A) \cap [2 c_{n}, 2 d_{n}] = W \cap [2 c_{n}, 2 d_{n}]$ .

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 $2 α$ . 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 $⌈ 2 α g - 1 ⌉$ arithmetic sequences [We call a set of the form $a + d ℕ$ an arithmetic sequence of difference $d$ and call a set of the form ${a, a + d, a + 2 d, ..., a + k d}$ 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 $α$ , 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 $(A + A) \cap [2 c_{n}, 2 d_{n}] = W \cap [2 c_{n}, 2 d_{n}]$ .

MR Zbl

DOI : 10.5802/jtnb.547

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

Affiliations des auteurs :

Prerna Bihani ¹ ; Renling Jin 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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     author = {Prerna Bihani and Renling Jin},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.547/}
}

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Prerna Bihani; Renling Jin. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.547/

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