Diophantine approximation with prime variables and mixed powers

Abstract

Assume that $\lambda _1,\ldots ,\lambda _4$ are positive real numbers with $\lambda _1/\lambda _2$ irrational and algebraic. Let $\mathcal {V}$ be a well-spaced sequence and $\updelta >0$. For any given positive integer $k \ge 4$ and any $\varepsilon >0$, we improve the upper bound of the number of $\upsilon \in \mathcal {V}$ with $\upsilon \le X$ for which the inequality $\vert \lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^4+\lambda _4p_4^k-\upsilon \vert < \upsilon ^{-\updelta }$ has no solution in primes $p_1,\dots ,p_4$.