  We show that for infinitely many primes $p$, there exist dual functions of
order $k$ over $\mathbb{F}_p^n$ that cannot be approximated in
$L_\infty$-distance by polynomial phase functions of degree $k-1$. This answers
in the negative a natural finite-field analog of a problem of Frantzikinakis on
$L_\infty$-approximations of dual functions over $\mathbb{N}$ (a.k.a. multiple
correlation sequences) by nilsequences.
