Diophantine inequality involving binary forms

Abstract

Let d ⩾ 3 be an integer, and set r = 2^d−1 + 1 for 3 ⩽ d ⩽ 4, $\tfrac{{17}} {{32}} \cdot 2^d + 1$ for 5 ⩽ d ⩽ 6, r = d²+d+1 for 7 ⩽ d ⩽ 8, and r = d²+d+2 for d ⩾ 9, respectively. Suppose that Φ _i (x, y) ∈ ℤ[x, y] (1 ⩽ i ⩽ r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2−d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.