Kneser’s theorem for upper Banach density

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 (2006)

  • Volume: 18, Issue: 2, page 323-343
  • ISSN: 1246-7405

Abstract

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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 ⊆ W 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 ) ∩ [ 2 c n , 2 d n ] = W ∩ [ 2 c n , 2 d n ] .

How to cite

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Bihani, Prerna, and Jin, Renling. "Kneser’s theorem for upper Banach density." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 323-343. <http://eudml.org/doc/249642>.

@article{Bihani2006,
abstract = {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 $\lbrace a, a+d, a+2d,\ldots ,a+kd\rbrace $ 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\rightarrow \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 $(d_n-c_n)/(b_n-a_n)\rightarrow 1$ and $(A+A)\cap [2c_n,2d_n]=W\cap [2c_n,2d_n]$.},
affiliation = {Department of Mathematics University of Notre Dame Notre Dame, IN 46556, U.S.A.; Department of Mathematics College of Charleston Charleston, SC 29424, U.S.A.},
author = {Bihani, Prerna, Jin, Renling},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Upper Banach density; inverse problem; nonstandard analysis; upper Banach density},
language = {eng},
number = {2},
pages = {323-343},
publisher = {Université Bordeaux 1},
title = {Kneser’s theorem for upper Banach density},
url = {http://eudml.org/doc/249642},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Bihani, Prerna
AU - Jin, Renling
TI - Kneser’s theorem for upper Banach density
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 323
EP - 343
AB - 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 $\lbrace a, a+d, a+2d,\ldots ,a+kd\rbrace $ 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\rightarrow \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 $(d_n-c_n)/(b_n-a_n)\rightarrow 1$ and $(A+A)\cap [2c_n,2d_n]=W\cap [2c_n,2d_n]$.
LA - eng
KW - Upper Banach density; inverse problem; nonstandard analysis; upper Banach density
UR - http://eudml.org/doc/249642
ER -

References

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  8. R. Jin, Inverse problem for upper asymptotic density. The Transactions of American Mathematical Society 355 (2003), No. 1, 57–78. Zbl1077.11007MR1928077
  9. R. Jin, Solution to the Inverse problem for upper asymptotic density. Journal für die reine und angewandte Mathematik 595 (2006), 121–166. Zbl1138.11045MR2244800
  10. T. Lindstrom, An invitation to nonstandard analysis in Nonstandard Analysis and Its Application. Ed. by N. Cutland. Cambridge University Press, 1988. Zbl0658.03044MR971064
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