Characterizing the structure of \(A+B\) when \(A+B\) has small upper Banach density
type: Item
instance of: scholarly article
title: Characterizing the structure of \(A+B\) when \(A+B\) has small upper Banach density
zbMATH Open document ID: 1201.11016
author: Renling Jin
published in: Journal of Number Theory
publication date: 2010-07-07T00:00:00Z
review text: For a set \(A\) of non-negative integers let \(BD(A)\) its upper Banach density and \(A(a,b)\) the number of elements in the interval \([a,b]\). The author proves: Theorem. Let \(A,B\subseteq {\mathbb N}\) be such that \(BD(A)=\alpha\), \(BD(B)=\beta\), and \(BD(A+B)<\alpha+\beta\). Then there are positive \(g\in{\mathbb N}\) and \(G\subseteq [0,g-1]\) such that: 1. \(BD(A+B)\geq \alpha+\beta-\frac{1}{g}\), 2. \(A+B\subseteq G+g{\mathbb N}\), 3. if \(\{[a_n^{(i)},b_n^{(i)}]:n\in{\mathbb N}\}\) for \(i=1,2\) are two sequences of intervals such that \[ \lim_{n\to\infty}(b_n^{(i)}-a_n^{(i)})=\infty\quad \text{for}\,i=1,2, \] \[ \lim_{n\to\infty}\frac{A(a_n^{(1)},b_n^{(1)})}{b_n^{(1)}-a_n^{(1)}+1}=\alpha, \lim_{n\to\infty}\frac{B(a_n^{(2)},b_n^{(2)})}{b_n^{(2)}-a_n^{(2)}+1}=\beta, \] and \[ 0<\liminf_{n\to\infty}\frac{b_n^{(1)}-a_n^{(1)}}{b_n^{(2)}-a_n^{(2)}}\leq \limsup_{n\to\infty}\frac{b_n^{(1)}-a_n^{(1)}}{b_n^{(2)}-a_n^{(2)}}<\infty, \] then there exist \([c_n^{(i)},d_n^{(i)}]\subseteq[a_n^{(i)},b_n^{(i)}]\) for each \(n\in{\mathbb N}\) and \(i=1,2\) such that \[ \lim_{n\to\infty}\frac{d_n^{(i)}-c_n^{(i)}}{b_n^{(i)}-a_n^{(i)}}=1 \] and \[ (A+B)\cap [c_n^{(1)}+c_n^{(2)},d_n^{(1)}+d_n^{(2)}]=(G+g{\mathbb N})\cap [c_n^{(1)}+c_n^{(2)},d_n^{(1)}+d_n^{(2)}]. \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 5731635
zbMATH Keywords: upper Banach density
zbMATH Keywords: sum sets
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.02.008
OpenAlex ID: W2025095324
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DOI: 10.1016/J.JNT.2010.02.008
description: scientific article; zbMATH DE number 5731635