Large values of the Gowers-Host-Kra seminorms

The Gowers uniformity norms ∥f∥ U k (G) of a function f: G → C on a finite additive group G, together with the slight variant ∥f∥ U k ([N]) defined for functions on a discrete interval [N]:= {1, ...,N}, are of importance in the modern theory of counting additive patterns (such as arithmetic progressions) inside large sets. Closely related to these norms are the Gowers-Host-Kra seminorms ∥f∥ U k (X) of a measurable function f: X → C on a measure-preserving system X = (X,X,µ, T). Much recent effort has been devoted to the question of obtaining necessary and sufficient conditions for these Gowers norms to have non-trivial size (e.g., at least η for some small η > 0), leading in particular to the inverse conjecture for the Gowers norms and to the Host-Kra classification of characteristic factors for the Gowers-Host-Kra seminorms.

In this paper, we investigate the near-extremal (or "property testing") version of this question, when the Gowers norm or Gowers-Host-Kra seminorm of a function is almost as large as it can be, subject to an L ∞ or L p bound on its magnitude. Our main results assert, roughly speaking, that this occurs if and only if f behaves like a polynomial phase, possibly localised to a subgroup of the domain; these results can be viewed as higher-order analogues of a classical result of Russo [29] and Fournier [10], and are also related to the polynomiality testing results over finite fields of Blum-Luby-Rubinfeld [6] and Alon-Kaufman-Krivelevich-Litsyn-Ron [1]. We investigate the situation further for the U 3 norms, which are associated to 2-step nilsequences, and find that there is a threshold behaviour, in that non-trivial 2-step nilsequences (not associated with linear or quadratic phases) only emerge once the U 3 norm is at most 2−1/8 of the L ∞ norm.