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On a ternary Diophantine inequality

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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.

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Acknowledgments

This work was written whilst the author was visiting Texas State University. The author is very grateful to Prof. Xingde Jia for his encouragement and comments. The author would also like to thank Texas State for the pleasant working environment. The author also extends his special gratitude to the reviewer for his/her careful reading and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Weili Yao

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  1. Weili Yao
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Correspondence to Weili Yao.

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This work is supported by NSFC (Grant Nos.11301325 and 61201258) and the China Scholarship Council (20133050).

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Yao, W. On a ternary Diophantine inequality. Ramanujan J 44, 155–175 (2017). https://doi.org/10.1007/s11139-016-9815-z

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  • DOI: https://doi.org/10.1007/s11139-016-9815-z

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