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.

Higher-order Fourier analysis, Dual functions, Finite fields, Additive combinatorics, Coding theory
12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Algorithms and Complexity

Briët, J, & Labib, F.S. (2021). High-entropy dual functions over finite fields and locally decodable codes.