Logarithmic Bounds for Translation-Invariant Equations in Squares
Logarithmic Bounds for Translation-Invariant Equations in Squares
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 …