Diophantine inequalities with mixed powers

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 $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ are rational. A given sequence of positive integers {n_i}_{i = 1}^s is said to have property (P) (($\text{P}{}^1$) 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}^sλ_ix_iⁿⁱ| < (max x_i)^?σ has infinitely many solutions in positive integers (primes respectively) x₁, x₂,…, 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 ($\text{P}{}^1$)), is that Σ_{i = 1}^∞n_i^?1 = ∞. These are parallel to some striking results of G. A. Fre?man, E. J. Scourfield and K. Thanigasalam.