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Nov 21, 2017 at 12:03 comment added Will Sawin It seems that we are allowed to choose $C$. Then it is easy to choose $C,D$ as long as $k\ell \leq p-1$, by choosing $C$ to be an interval of length $k$ in the multiplicative group and $D$ to consist of muliples of $k$. So in fact we win for $k \leq \sqrt{p-1}$.
Nov 21, 2017 at 7:14 history answered Greg Martin CC BY-SA 3.0