Large values of the Gowers-Host-Kra seminorms

The \emph{Gowers uniformity norms} $\|f\|_{U^k(G)}$ of a function f: G \to \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 \emph{Gowers-Host-Kra seminorms} $\|f\|_{U^k(X)}$ of a measurable function f: X \to \C on a measure-preserving system $X = (X, {\mathcal X}, \mu, T)$. 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 $\eta$ for some small $\eta > 0$), 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 $L^\infty$ 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; this can be viewed as a higher-order analogue of classical results of Russo and Fournier, and are also related to the polynomiality testing results over finite fields of Blum-Luby-Rubinfeld and Alon-Kaufman-Krivelevich-Litsyn-Ron. 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 $L^\infty$ norm.Comment: 52 pages, no figures, to appear, Journal d'Analyse Jerusalem. This is the final version, incorporating the referee's suggestion

Automatic sequences as good weights for ergodic theorems

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$.Comment: 31 page