We show that for infinitely many primes p, there exist dual functions of order k over Fnp that cannot be approximated in L∞-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∞-approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences.

Forum of Mathematics, Sigma
Algorithms and Complexity

Briët, J., & Labib, F. (2021). High-entropy dual functions over finite fields and locally decodable codes. Forum of Mathematics, Sigma, 9(e19). doi:10.1017/fms.2021.1