Small fractional parts of additive forms (Q1825235)

Let \(f(x)=\theta_ 1x^ k_ 1+...+\theta_ sx^ k_ s\) be an additive form where all \(\theta_ j\) are algebraic. For any \(\alpha\in {\mathbb{R}}\), \(\| \alpha \|\) denotes the distance from \(\alpha\) to the nearest integer. In this paper the author adapts \textit{D. R. Heath- Brown}'s results on Weyl's inequality [J. Lond. Math. Soc., II. Ser. 38, 216-230 (1988; Zbl 0619.10046); Mathematika 35, 28-37 (1988; Zbl 0629.10029)] to prove that if \(s=4k\) and \(N\geq N(k,\theta)\) then the inequality \(\| f(x)\| \leq N^{-1+(2/e)}\) has a solution x with \(1\leq x_ i\leq N\). The author also obtains that if \(k\geq 6\), \(s\leq 2^{k-3}\) then for any \(\epsilon >0\) the inequalities \(\| f(x)\| \leq N^{-(4s/3K)+\epsilon}\) has a solution x with \(1\leq x_ i\leq N\) where \(N\geq N(\epsilon,k,\theta)\) and \(K=2^{k-1}\). This generalizes some localized results due to Heath-Brown [ibid. p. 30] for \(s=1\).