A Host–Kra $$\mathbb {F}_2^\omega $$-system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the $$U^6(\mathbb {F}_2^n)$$ norm

Abstract It was conjectured by Bergelson et al. (Geom Funct Anal 19(6):1539–1596, 2010) that every Host–Kra $$\mathbb {F}_p^\omega $$ F p ω -system of order k is an Abramov system of order k . This conjecture has been verified for $$k \le p+1$$ k ≤ p + 1 . In this paper we show that the conjecture fails when $$k=5, p=2$$ k = 5 , p = 2 . We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $$f: \mathbb {F}_2^n \rightarrow \mathbb {C}$$ f : F 2 n → C of large Gowers norm $$\Vert f\Vert _{U^6(\mathbb {F}_2^n)}$$ ‖ f ‖ U 6 ( F 2 n ) which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial e ( P ), but with the property that all such phase polynomials e ( P ) are “non-measurable” in the sense that they cannot be well approximated by functions of a bounded number of random translates of f . A simpler version of our construction can also be used to answer a question of Candela et al. (Ergodic Theory Dyn Syst 43(12):3971–4040, 2023).