Characterizing the structure of \(A+B\) when \(A+B\) has small upper Banach density
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 \[ (A+B)\cap [c_n^{(1)}+c_n^{(2)},d_n^{(1)}+d_n^{(2)}]=(G+g{\mathbb N})\cap [c_n^{(1)}+c_n^{(2)},d_n^{(1)}+d_n^{(2)}]. \]
scientific article; zbMATH DE number 5731635