A diophantine inequality. II (Q1192382)

The authors prove the following theorem: Let \(\lambda_ 1,\dots,\lambda_ 5\) be non-zero real numbers and not all in rational ratios and not all of the same sign. Then, given a real number \(\eta\), the inequality \[ |\lambda_ 1 x_ 1^ 2+\dots+\lambda_ 5 x_ 5^ 2+\eta|<(\max_{1\leq i\leq 5} | x_ i|)^{-{1\over 2}+\delta} \] has infinitely many solutions in positive integers \(x_ 1,\dots,x_ 5\) for any \(\delta>0\). This improves the result of Part I [cf. Acta Math. Sin. 30, 598-604 (1987; Zbl 0635.10016)] where the exponent on the right hand side \(-1/3+\varepsilon\) was obtained.