The subject matter of the paper is investigation of different properties of positive integers \(n\) for which \([n\alpha+s]\neq[n\beta+s].\) Let \(N(\beta,\alpha,s)\) be the set of all such numbers and \(\Psi(\beta,\alpha,s)\) be the least such integer. Let \(\Psi_k(\beta,\alpha,s)\) be the k-th integer \(n\) for which \([n\alpha+s]\neq[n\beta+s]\). First of all, the asymptotic formula for the probability \(P(\Psi(\beta,\alpha,0)>Q)\) is obtained. The main idea of the proof is to reduce the problem to investigating the sums of the form \(\sum_{j}(\gamma_{j+1}-\gamma_j)^2\) where \(\gamma_j\) is a Farey fraction. Also the formula for \(P(\Psi(\beta,\alpha,0)=N)\) is obtained. With the use of this formula the expected value of \(\Psi\) is determined. Applying much more complicated methods (again the key ingredient is Farey fractions) the authors prove the asymptotic formula for the probability \(P(\Psi_2(\beta,\alpha,0)>Q).\)
Secondly, the authors characterize the structure of the set \(N(\beta,\alpha,s).\) To do this the so called semiconvergents sequences of approximating fractions constructed in the previous paper of the first author are used. Using the information about the set \(N(\beta,\alpha,s)\) a theoretical formula for \(\Psi_k(a/b,c/d,s)\) is obtained.