LAUSR

AbstractLet d ⩾ 3 be an integer, and set r = 2d−1 + 1 for 3 ⩽ d ⩽ 4, $$\tfrac{{17}} {{32}} \cdot 2^d + 1$$1732⋅2d+1 for 5 ⩽ d ⩽ 6, r = d2+d+1 for 7 ⩽ d ⩽ 8, and r = d2+d+2 for d ⩾ 9, respectively. Suppose that Φi(x, y) ∈ ℤ[x, y] (1 ⩽ i ⩽ r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,..., λr are nonzero real numbers with λ1/λ2 irrational, and λ1Φ1(x1, y1) + λ2Φ2(x2, y2) + · · · + λrΦr(xr, yr) is indefinite. Then for any given real η and σ with 0 < σ < 22−d, it is proved that the inequality $$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$|∑i=1rλiΦ(xi,yi)i+η|<(max{|xi|,|yi|}1≤i≤r)−σ has infinitely many solutions in integers x1, x2,..., xr, y1, y2,..., yr. This result constitutes an improvement upon that of B. Q. Xue.