Solution-free sets for sums of binary forms

TL;DR

This paper studies the density of subsets of the 2D integer lattice that avoid non-trivial solutions to additive equations involving binary forms, introducing a Vinogradov mean value theorem analogue for binary forms.

Contribution

It provides new quantitative bounds on the density of solution-free sets and develops a Vinogradov mean value theorem analogue for binary forms.

Findings

Quantitative estimates for solution-free sets density

Development of a Vinogradov mean value theorem analogue for binary forms

New bounds on additive equations involving binary forms

Abstract

We obtain quantitative estimates for the asymptotic density of subsets of the two-dimensional integer lattice which contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov's mean value theorem applicable to binary forms.