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).