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Abstract : 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 Z/NZ introduced by Gowers in his proof of Szemeredi'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 Szemeredi's Theorem) defined by the authors. For each integer k >= 1, we define seminorms on l(infinity)(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.
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Submitted on : Tuesday, May 1, 2012 - 7:45:34 PM
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  • HAL Id : hal-00693050, version 1


Bernard Host, Bryna Kra. UNIFORMITY SEMINORMS ON l(infinity) AND APPLICATIONS. Journal d'analyse mathématique, Springer, 2009, 108 (?), pp.219--276. ⟨hal-00693050⟩



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