Small fractional parts of binary forms

TL;DR

This paper establishes new bounds on the fractional parts of certain binary forms by leveraging recent advances in Vinogradov's mean value theorem and exponential sums, improving previous estimates for the minimal fractional parts.

Contribution

It introduces improved bounds on fractional parts of binary forms using recent progress in exponential sum estimates and Vinogradov's mean value theorem.

Findings

Derived superior bounds for fractional parts depending on form parameters

Utilized recent progress in Vinogradov's mean value theorem

Enhanced previous estimates for the minimal fractional parts

Abstract

We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with $\alpha_k,\alpha_l,\ldots,\alpha_0\in\mathbb{R}$ 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 \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}.\end{equation*}