On a ternary Diophantine inequality

Abstract

Let \(\Omega (n)\) be the total number of prime factors of n, and let \(\lambda _j\) be the real numbers satisfying suitable conditions. Let \(J_k(N)\) denote the number of solutions to the inequality $$\begin{aligned} \left\{ \begin{array}{ll} |\lambda _1n_1+\lambda _2n_2+\lambda _3n_3+\eta |<\varepsilon ,&{} \\ \Omega (n_j)= k,&{} \\ 2\le n_j\le N \quad (j=1,2,3).&{} \end{array} \right. \end{aligned}$$ In this note, we investigate the properties of \(J_k(N)\) for any integer \(k\ge 1\), which is allowed to tend to infinity with respect to N. Using an asymptotic formula for the weighted exponential sums, we obtain a sharper lower bound for it and also discuss an application of the main result.