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 $|A| \, \ll \, N^{1-\frac1k\min\{s \, σ_k, \, 1/2\}+ε}$ for any fixed $ε>0$, where $σ_k >0 $ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take $σ_k=\max\left\{2^{1-k}, \, \frac{1}{k(k-1)}\right\}$ using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set $A\subset[N]$ with $|A|\gg N^{1-s/k}$ for which $A-A$ contains no sum of $s$-many positive $k$-th powers, so our power-saving bound is of the correct shape.