Uniformity seminorms on $\ell^\infty$ and applications
| dc.creator | Kra, Bryna | |
| dc.creator | Host, Bernard | |
| dc.date | 2007-11-22 | |
| dc.date.accessioned | 2026-07-07T08:44:36Z | |
| dc.date.available | 2026-07-07T08:44:36Z | |
| dc.description | 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 $\Z/N\Z$ 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 $\ell^\infty(\Z)$ 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. | |
| dc.identifier | https://arxiv.org/abs/0711.3637 | |
| dc.identifier | http://arxiv.org/abs/0711.3637 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/142714 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 37A45 | |
| dc.title | Uniformity seminorms on $\ell^\infty$ and applications | |
| dc.type | text |