The Furstenberg-Sárközy theorem for sums of an even number of odd powers

We obtain a Furstenberg-Sárközy-type result for sets $A\subset [N]$ whose difference set $A-A$ does not contain the sum of $s$-many $k$-th powers of positive integers, with $k>1$ odd and $s>0$ even. Namely, we prove that such sets must satisfy a power-saving bound $\mathrm{\mid }A\mathrm{\mid }\ll N^{1-\frac{1}{k}\min \{s{\sigma }_k,1\mathrm{/}2\}+\epsilon }$ for any fixed $\epsilon >0$, where ${\sigma }_k>0$ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take ${\sigma }_k=\max \{2^{1-k},\frac{1}{k(k-1)}\}$ using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set $A\subset [N]$ with $\mathrm{\mid }A\mathrm{\mid }\gg N^{1-s\mathrm{/}k}$ for which $A-A$ contains no sum of $s$-many positive $k$-th powers, so our power-saving bound is of the correct shape.