Croftian sequences (Q1175248)

The authors study the following and related problems: Characterize sequences \(\{c_ n\}\to\infty\) which have the following property: If \(f:\mathbb{R}\to\mathbb{R}\) is continuous and \(\lim_{n\to\infty}f(x+c_ n)=0\) for all real \(x\), then \(\lim_{x\to\infty}f(x)=0\). For related results see, e.g., \textit{N. H. Bingham, C. M. Goldie} and \textit{J. L. Teugels}: ``Regular variation'' (1987; Zbl 0617.26001), pp. 49-51, and the references cited therein. It is, e.g., shown that the following statements are equivalent: 1) \(\{c_ n\}\) does not have the desired property, 2) there exists some sequence \(\{a_ k\}\to\infty\) such that \(\{a_ k-c_ n;\;k,n\in\mathbb{N}\}\) is nowhere dense, 3) either \(c_{n+1}- c_ n\to\infty\) or there exists a subsequence \(\{n_ k\}\) such that \(\{c_{n_ k}-c_ n;\;k,n\in\mathbb{N}\}\) is nowhere dense.