On a diagonal quadric in dense variables

TL;DR

This paper investigates the solvability of diagonal quadratic equations in dense subsets of integers, providing quantitative bounds and employing advanced analytic methods like the circle method and Roth's density increment argument.

Contribution

It introduces a restriction theory approach to handle equations with at least 7 variables, advancing the understanding of quadratic solutions in dense integer sets.

Findings

Quantitative bounds on the size of sets with no non-trivial solutions

Application of the circle method and Roth's density increment argument

Handling equations in s ≥ 7 variables

Abstract

We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets A \subset Z and show quantitative bounds on the size of A if there are no non-trivial solutions. We use the circle method and Roth's density increment argument. Due to a restriction theory approach we can deal with equations in s \geq 7 variables.