On the uniformity of some sequences of rational numbers

Abstract. Let \({\cal S}\) be a subset of rational numbers. For \(x\geqslant 1\) we introduce the set \({\cal S}_x\), \({\cal S}_x \subset {\cal S}\), which consists of numbers \(m/n\in {\cal S}, (m,n)=1, n\leqslant x\). For \(J=(\lambda_1;\lambda_2)\), \(J\subset (0;+\infty )\), we denote \(|J|=\lambda_2-\lambda_1\), \(J^u=(\lambda_1; \lambda_1+u|J|)\), and \(F_x(u)=\#({\cal S}_x\cap J^u)/\#({\cal S}_x\cap J)\), where \(0\leqslant u\leqslant 1\). The discrepancy \(\sup_{u}|F_x(u)-u|\) is evaluated for some subsets \({\cal S}\), specified by arithmetical conditions.