Logarithmic bounds for translation-invariant equations in squares

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 $s\ge 7$ and the coefficients ${\lambda }_i$ sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form $(\log \log N{)}^{-c}$.

arXiv comment19 pages, fixed typos, improved notation