Diophantine approximation with four squares and one kth power of primes

Abstract

Let k be an integer with $k\ge 3$ and $\eta$ be any real number. Suppose that $\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu$ are non-zero real numbers, not all of the same sign and $\lambda _1/\lambda _2$ is irrational. It is proved that the inequality $|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }$ has infinitely many solutions in prime variables $p_1, p_2, \ldots , p_5$, where $0<\sigma <\frac{1}{16}$ for $k=3,\ 0<\sigma <\frac{5}{3k2^k}$ for $4\le k\le 5$ and $0<\sigma <\frac{40}{21k2^k}$ for $k\ge 6$. This gives an improvement of an earlier result.