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