On inhomogeneous approximations to irrational numbers with arithmetical conditions (Q1349485)

Let \(\xi\) be an irrational number. Then for every real number \(\eta\) and integers \(a,b,s\) \((1\leq a,b\leq s)\) and for every \(\varepsilon>0\) there are infinitely many pairs of integers \(P\) and \(Q>0\) such that \[ |Q\xi-P- \eta|<{(1+ \varepsilon) \cdot s^2\over \sqrt 5\cdot Q} \] where \(P\equiv a \pmod s\), \(Q\equiv b \pmod s\). If \(\xi,\eta,a,b,s\) satisfy special conditions, then it is possible to improve this inequality.