An improved estimate for certain Diophantine inequalities

Abstract

Let λ 1 , … , λ 8 {\lambda _1}, \ldots ,{\lambda _8} be any nonzero real numbers such that not all λ j {\lambda _j} are of the same sign and not all ratios λ j / λ k {\lambda _j}/{\lambda _k} are rational. If η , α \eta ,\alpha are any real numbers with 0 > α > 3 / 70 0 > \alpha > 3/70 then | η + Σ j = 1 8 λ j n j 3 | > ( max n j ) − α |\eta + \Sigma _{j = 1}^8{\lambda _j}n_j^3| > {(\max {n_j})^{ - \alpha }} has infinitely many solutions in positive integers n j {n_j} .