Multiple correlation sequences not approximable by nilsequences

Abstract

We show that there is a measure-preserving system $(X,\mathscr {B}, \mu , T)$ together with functions $F_0, F_1, F_2 \in L^{\infty }(\mu )$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int _X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 \, d\mu $ is not an approximate integral combination of $2$ -step nilsequences.

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

“…Remark 1.3. Ideas behind Theorem 1.2 recently inspired similar examples in the integer setting for 3-term progressions in joint work of the first author and Green [BG20].…”