Small fractional parts of binary forms. (English) Zbl 1533.11132
Summary: We obtain bounds on fractional parts of binary forms of the shape
\[
\Psi(x,y)=\alpha_kx^k+\alpha_lx^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k
\]
with \(\alpha_k,\alpha_l,\dots,\alpha_0\in\mathbb{R}\) and \(l\leq k-2\). By exploiting recent progress on Vinogradov’s mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent \(\sigma\), depending on \(k\) and \(l\), such that
\[
\min_{\substack{0\leq x,y\leq X\\ (x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}.
\]
MSC:
11J54 | Small fractional parts of polynomials and generalizations |
11E76 | Forms of degree higher than two |
11L07 | Estimates on exponential sums |