Diophantine Approximation with Mixed Powers of Primes

Abstract

Let $k$ be an integer with $k \geq 3$. Let $\lambda_1$, $\lambda_2$, $\lambda_3$ be non-zero real numbers, not all negative. Assume that $\lambda_1/\lambda_2$ is irrational and algebraic. Let $\mathcal{V}$ be a well-spaced sequence, and $\delta \gt 0$. In this paper, we prove that, for any $\varepsilon \gt 0$, the number of $\upsilon \in \mathcal{V}$ with $\upsilon \leq X$ such that the inequality \[ |\lambda_1 p_1^2 + \lambda_2 p_2^2 + \lambda_3 p_3^k - \upsilon| \lt \upsilon^{-\delta} \] has no solution in primes $p_1$, $p_2$, $p_3$ does not exceed $O(X^{1-2/(7m_2(k))+2\delta+\varepsilon})$, where $m_2(k)$ relies on $k$. This refines a recent result. Furthermore, we briefly describe how a similar method can refine a previous result on a Diophantine problem with two squares of primes, one cube of primes and one $k$-th power of primes.