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Uniformity seminorms on $\ell^{\infty}$ and applications. (English) Zbl 1183.37011
Summary: A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Bbb Z/N\Bbb Z$ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer $k \geq 1$, we define seminorms on $\ell^{\infty}(\Bbb Z)$ analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 37A05 Measure-preserving transformations 37A45 Relations of ergodic theory with number theory and harmonic analysis 20D15 Nilpotent finite groups, $p$-groups
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