Small fractional parts of binary forms

Abstract

We obtain bounds on fractional parts of binary forms of the shape $$Ψ(x,y)=α_k x^k+α_l x^ly^{k-l}+α_{l-1}x^{l-1}y^{k-l+1}+\cdots+α_0 y^k$$ with $α_k,α_l,\ldots,α_0\in\mathbb{R}$ 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 $σ$, depending on $k$ and $l,$ such that \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|Ψ(x,y)\|\leq X^{-σ+ε}.\end{equation*}