Diophantine inequalities with mixed powers

Abstract

Let {λi}i = 1 s (s ≥ 2) be a finite sequence of non-zero real numbers, not all of the same sign and in which not all the ratios λi λj are rational. A given sequence of positive integers {ni}i = 1 s is said to have property (P) ((P*) respectively) if for any {λi}i = 1 s and any real number η, there exists a positive constant σ, depending on {λi}i = 1 s and {ni}i = 1 s only, so that the inequality |η + Σi = 1 s λixi ni| < (max xi)-σ has infinitely many solutions in positive integers (primes respectively) x1, x2,..., xs. In this paper, we prove the following result: Given a sequence of positive integers {ni}i = 1 ∞, a necessary and sufficient condition that, for any positive integer j, there exists an integer s, depending on {ni}i = j ∞ only, such that {ni}i = j j + s - 1 has property (P) (or (P*)), is that Σi = 1 ∞ ni -1 = ∞. These are parallel to some striking results of G. A. Freǐman, E. J. Scourfield and K. Thanigasalam. © 1982.