Logarithmic bounds for translation-invariant equations in squares

Kevin Henriot

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 coeff…