Logarithmic bounds for translation-invariant equations in squares

TL;DR

This paper proves that certain quadratic equations with specific coefficient conditions have non-trivial solutions in dense subsets of integers, with bounds improving from double logarithmic to single logarithmic scale.

Contribution

It establishes new logarithmic density bounds for solutions to translation-invariant quadratic equations in integers, improving previous results.

Findings

Non-trivial solutions exist in subsets of density (log N)^{-c_s}

Results apply to equations with coefficients summing to zero and sign conditions

Improves bounds from (log log N)^{-c} to (log N)^{-c_s}

Abstract

We show that the equation $\lambda_1 n_1^2 + ... + \lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\log N)^{-c_s}$, provided that $s \geq 7$ and the coefficients $\lambda_i$ sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form $(\log\log N)^{-c}$.