Cofinite subsets of asymptotic bases for the positive integers
Let \(A\subseteq\mathbb Z\), \(| \{a\in A\mid a<0\}| <\infty\) and \(F\subseteq A\), \(| F| <\infty\). For \(h\in\mathbb N\) let \(hA:=\{\sum^h_{i=1} a_i\mid a_i\in A\}\). If there exists \(h\in\mathbb N\) such that \(hA\cap [n,\infty [=\mathbb N_0\cap [n,\infty [\) for some \(n\in\mathbb N_0\) then \(A\) is called an asymptotic basis. The smallest \(h\) with this property is denoted by \(g(A)\). Theorem 1 in this paper verifies a conjecture of the second author: If \(hA\) contains an infinite arithmetic progression with difference \(d:=\gcd \{a- a'\mid a,a'\in A\setminus F\}\), then there exists some \(h'\in\mathbb N\) such that \(h'(A\setminus F)\) contains an infinite arithmetic progression with difference \(d\). In particular, if \(A\) is an asymptotic basis, then \(A\setminus F\) is an asymptotic basis if and only if \(d=1\). This is a generalization of a result of \textit{P. Erdős} and \textit{R. L. Graham} [Acta Arith. 37, 201--207 (1980; Zbl 0443.10036)]. For given \(h,k\in\mathbb N\) let \(G_k(h):=\max_{g(A)\leq h}(\max_{F\subseteq A, | F| =k} g(A\setminus F))\). Theorem 4 states for \(h\geq 2\): \[ (h/(k+1))^{k+1}+O(h^k)\leq G_k(h)\leq (2/k!)h^{k+1}+O(h^k). \] Essential use of Kneser's addition theorem is made in the proof for the upper bound. For the estimate from below see the second author [Lect. Notes Math. 1052, 273--277 (1984; Zbl 0544.10058)].
scientific article; zbMATH DE number 3924916