Diophantine inequalities with mixed powers

Abstract

Let { λ i } 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 λ i λ j are rational. A given sequence of positive integers { n i } i = 1 s is said to have property ( P) (( P ∗ ) respectively) if for any { λ i } i = 1 s and any real number η, there exists a positive constant σ, depending on { λ i } i = 1 s and { n i } i = 1 s only, so that the inequality | η + Σ i = 1 s λ i x i ni | < (max x i ) − σ has infinitely many solutions in positive integers (primes respectively) x 1, x 2,…, x s . In this paper, we prove the following result: Given a sequence of positive integers { n i } i = 1 ∞, a necessary and sufficient condition that, for any positive integer j, there exists an integer s, depending on { n i } i = j ∞ only, such that { n i } i = j j + s − 1 has property ( P) (or ( P ∗ )), is that Σ i = 1 ∞ n i −1 = ∞. These are parallel to some striking results of G. A. Freĭman, E. J. Scourfield and K. Thanigasalam.