Let \(\lambda_1, \dots, \lambda_8\) be any numbers with all \(\lambda_j \geq 1\). In the paper, using a variant of the Hardy-Littlewood circle method, the author proves that for any \(\varepsilon>0\) there are integers \(x_1, \dots, x_8\) such that the inequalities \[ \left|\sum^8_{j=1} \lambda_j x_j^3 \right |<1 \quad \text{and} \quad 0<\sum^8_{j=1} \lambda_j|x_j |^3 \ll \left( \prod_{j=1}^8 \lambda_j \right)^{(15/8) + \varepsilon} \] hold simultaneously. He also claims without proof that the above exponent 15/8 can be replaced by 1 if the number of variables in the above inequalities is increased to 9.
Although the methods in the proof have its origins in a work of B. J. Birch and H. Davenport [Acta Math. 100, 259–279 (1958; Zbl 0082.26002)], the author’s previous results [J. Lond. Math. Soc., II. Ser. 50, 25–42 (1994; Zbl 0810.11014) and Sémin. Théor. Nombres, Paris, 1990–1991, Prog. Math. 108, 23–33 (1991; Zbl 0815.11040)] on a mean value estimate and on a bound for small integer solutions of a similar equation play important roles in his proof.
As a corollary the author also obtains that the inequality
\[ \left|\sum^8_{j=1} \lambda_jx_j^3 \right|< \varepsilon \]
has a solution in integers \(x_j\), not all zero, satisfying \(|x_j|\ll \varepsilon^{-\theta}\) where \(\theta>14/3\) and \(1> \varepsilon>0\).
[For Part I, see Mathematika 35, 51–58 (1988; Zbl 0659.10015).]