Integer-detecting sequences (Q1279791)

Let \(\|\alpha\|\) denote the distance from the real number \(\alpha\) to the nearest integer. Let \(\varepsilon>0\). A sequence \(\{m_n\}\) of positive integers is called integer-detecting of height \(\varepsilon\) if every rational number \(r\), such that \(\| m_nr\|<\varepsilon\) for all \(n\geq 0\) is an integer. A sequence of positive integers is called integer-detecting if it is integer-detecting of height \(\varepsilon\) for some \(\varepsilon>0\). The paper deals with several criteria of integer-detecting sequences.