High-entropy dual functions over finite fields and locally decodable codes

TL;DR

This paper demonstrates that for infinitely many primes, certain dual functions over finite fields cannot be closely approximated by lower-degree polynomial phase functions, highlighting limitations in finite-field analogs of classical correlation sequence approximations.

Contribution

It establishes the existence of dual functions over finite fields that defy approximation by polynomial phase functions, answering a key open problem in finite-field harmonic analysis.

Findings

Existence of dual functions not approximable by lower-degree polynomials

Negative answer to a finite-field analog of a problem by Frantzikinakis

Implications for the structure of multiple correlation sequences over finite fields

Abstract

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.