Approximation of \(\lfloor na+s \rfloor\) and the zero of \(\{ na+s \}\) (Q1343637)

Let \(\alpha\) and \(\beta\) be positive real numbers and \(s\) a real number satisfying \(0\leq s<1\). Let \(\lfloor x\rfloor\) denote the greatest integer \(\leq x\), and \(\{x\}= x- \lfloor x\rfloor\). Define \(\Psi (\alpha, \beta; s)\) to be the least positive integer \(n\) such that \(\lfloor n\alpha+s \rfloor\neq \lfloor n\beta+ s\rfloor\). When \(s=0\), a simple explicit formula for \(\Psi\) is given, and otherwise more complicated formulas are obtained. When \(\alpha\) is irrational, a formula for finding the least \(n\in \mathbb{Z}^ +\) such that \(\{n\alpha+ s\}=0\) is presented. A natural characterization of the approximation properties of intermediate convergents (including convergents) without reference to the apparatus of continued fractions is given. A new characterization of the sequence \(\lfloor n\alpha \rfloor\) for \(n\geq 1\) is found, and \(\lfloor n\alpha+s \rfloor\) for \(n\geq 1\) is also characterized in a different way.