A note on Diophantine approximation with prime variables and mixed powers

Abstract

Let \(k\ge 4\) be an integer. Suppose that \(\lambda _1,\lambda _2,\lambda _3,\lambda _4\) are positive real numbers, \(\frac{\lambda _1}{\lambda _2}\) is irrational and algebraic. Let \(\mathcal {V}\) be a well-spaced sequence, and \(\delta >0\). In this paper, we prove that, for any \(\varepsilon >0\), the number of \(\upsilon \in \mathcal {V}\) with \(\upsilon \le X\) such that the inequality

$$\begin{aligned} |\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^4+\lambda _4p_4^k-\upsilon |<\upsilon ^{-\delta } \end{aligned}$$

has no solution in primes \(p_1,p_2,p_3,p_4\) does not exceed \(O(X^{1-\sigma ^*(k)+2\delta +\varepsilon })\), where \(\sigma ^*(k)\) relies on k. This improves a recent result of Qu and Zeng (Ramanujan J 52:625–639, 2020).

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Acknowledgements

The author would like to thank the referee for many useful suggestions on the manuscript.

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Correspondence to Huafeng Liu.

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This work is supported by the National Natural Science Foundation of China (Grant No. 11801328).

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Liu, H. A note on Diophantine approximation with prime variables and mixed powers. Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00347-x

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Keywords

  • Diophantine inequality
  • Exceptional set
  • Sieve functions

Mathematics Subject Classification

  • 11P32
  • 11D75