Uniformity seminorms on $\ell^\infty$ and applications

TL;DR

This paper introduces new seminorms on f8(\u00bb) inspired by Gowers norms, analyzes their properties, and applies them to derive ergodic theorems involving nilsequences, advancing the understanding of patterns in integers.

Contribution

The paper defines and studies a family of seminorms on f8(bb) analogous to Gowers norms, and establishes their correlation properties and inverse theorems, with applications to ergodic theory.

Findings

Defined seminorms on f8(bb) similar to Gowers norms.

Proved correlation bounds with nilsequences using dual norms.

Established inverse theorems linking bounded sequences to nilsequences.

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 $\Z/N\Z$ introduced by Gowers in his proof of Szemer\'edi'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 Szemer\'edi'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.