Small fractional parts of polynomials

TL;DR

This paper leverages recent advances in Vinogradov's mean value theorem to derive new results on the small fractional parts of polynomials and additive forms, improving previous bounds and understanding.

Contribution

It introduces novel results on small fractional parts of polynomials using Bourgain, Demeter, and Guth's recent theorem, advancing prior research.

Findings

Improved bounds on fractional parts of polynomials

Enhanced understanding of additive forms

Refined estimates compared to earlier work

Abstract

Using the recent result of Bourgain, Demeter and Guth on Vinogradov's mean value, a number of new results about small fractional parts of polynomials and fractional parts of additive forms are obtained. These improve work of Baker, Cook, Danicic, Vaughan and Wooley.