Small fractional parts of binary forms

Abstract

We obtain bounds on fractional parts of binary forms of the shape $$\Psi (x,y)={\alpha }_k{x}^k+{\alpha }_l{x}^l{y}^{k-l}+{\alpha }_{l-1}{x}^{l-1}{y}^{k-l+1}+\cdots +{\alpha }_0{y}^k$$ with ${\alpha }_k,{\alpha }_l,\dots ,{\alpha }_0\in \mathbb{R}$ and $l\le k-2.$ 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 \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|Ψ(x,y)\|\leq X^{-σ+ε}.\end{equation*}

30 pages, all comments welcome