Diophantine approximation with mixed powers of primes

Abstract

Let k be a positive integer with $k \ge 5$, $\lambda _1, \lambda _2, \lambda _3, \lambda _4$ be nonzero real numbers, not all of the same sign, with $\lambda _1/\lambda _2$ irrational and algebraic. Suppose that ${\mathcal {V}}$ is a well-spaced sequence and $\delta >0$. By $E_{4k}({\mathcal {V}}, X, \delta )$, we denote the number of $v \in {\mathcal {V}}$ with $v \le X$ for which

$$\begin{aligned} |\lambda _1 p_1^2 + \lambda _2 p_2^2+ \lambda _3 p_3^4+ \lambda _4 p_4^k - v| < v^{-\delta } \end{aligned}$$

has no solution in primes $p_1,p_2,p_3,p_4$. In this paper, it is proved that $E_{4k}({\mathcal {V}}, X, \delta ) \ll X^{1-\sigma (k)+2\delta +\varepsilon }$, where $\sigma (k)$ relies on k. This result constitutes a refinement upon that of Qu and Zeng (Diophantine approximation with prime variables and mixed powers. Ramanujan J 52:625–639, 2020).