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We show that the equation   $\lambda _1 n_1^2 + \dotsb + \lambda _s n_s^2 = 0$   admits nontrivial solutions in any subset of   $[N]$   of density   $(\log N)^{-c_s}$  , provided that   $s \geq 7$   and the coefficients   $\lambda _i \in \mathbb {Z} \smallsetminus \{0\}$   sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form   $(\log \log N)^{-c}$  .

Logarithmic Bounds for Translation-Invariant Equations in Squares