Lethbridge Number Theory and Combinatorics Seminar: Kevin Henriot

We discuss the solvability of certain linear equations in sparse subsets of the squares. Specifically, we investigate equations of the form \begin{align*}     \lambda_1 n_1^2 + \dotsb + \lambda_s n_s^2 = 0, \end{align*} where $s \geq 7$ and the coefficients $\lambda_i$ sum to zero and satisfy certain sign conditions. We show that such equations admit non-trivial solutions in any subset of $[N]$ of density $(\log N)^{-c_s}$, improving upon the previous best of $(\log\log N)^{-c}$.