## 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

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

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### Cite this article

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