A note on Diophantine approximation with four squares and one k-th power of primes

Let \(\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu \) be non-zero real numbers, not all negative, with \(\lambda _1/\lambda _2\) irrational. Suppose that \(k \geqslant 3\) be an integer and \(\eta \) be any given real number. In this paper, it is proved that for any real number \(\sigma \) with \(0<\sigma <\frac{1}{\vartheta (k)}\), the inequality

$$\begin{aligned} |\lambda _1 p_1^2 + \lambda _2 p_2^2+ \lambda _3 p_3^2+ \lambda _4 p_4^2 + \mu p_5^k + \eta | < \left( \max \limits _{1\leqslant j \leqslant 5}p_j\right) ^{-\sigma } \end{aligned}$$

has infinitely many solutions in prime variables \(p_1,\cdots ,p_5\), where \(\vartheta (k) = \frac{32}{5}\lceil {\big (\frac{k}{2} + 1 - [\frac{k}{2}]\big )2^{[\frac{k}{2}]-1}}\rceil \) for \(3\leqslant k \leqslant 9\) and \(\vartheta (k) = \frac{32}{5}\lceil {\big (\frac{k}{2} - \frac{1}{2}[\frac{k}{2}]\big )\big ([\frac{k}{2}]+1\big )}\rceil \) for \(k \geqslant 10\). This result constitutes an improvement upon that of Q. W. Mu, M. H. Zhu and P. Li [13].