https://hal-upec-upem.archives-ouvertes.fr/hal-00693050Host, BernardBernardHostLAMA - Laboratoire d'Analyse et de Mathématiques Appliquées - UPEM - Université Paris-Est Marne-la-Vallée - Fédération de Recherche Bézout - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche ScientifiqueKra, BrynaBrynaKraUNIFORMITY SEMINORMS ON l(infinity) AND APPLICATIONSHAL CCSD2009[MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]Lama, Admin2012-05-01 19:45:342022-01-15 04:07:112012-05-01 19:46:07enJournal articles1A 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.