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

Abstract: 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 $k&gt;1$ odd and $s&gt;0$ 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 $ε&gt;0$, where $σ_k &gt;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 $|A|\gg N<sup>{1-s/k}$ for which $A-A$ contains no sum of $s$-many positive $k$-th powers, so our power-saving bound is of the correct shape.