Furstenberg systems of certain sequences of superpolynomial growth

TL;DR

This work addresses the ergodic-modeling of sequences built from intermediate-growth iterates, notably $G(x)=x^{\log^c x}$ with $0<c<1/2$, by analyzing their Furstenberg systems. The authors develop strong exponential-sum estimates and multidimensional equidistribution results, enabling a complete Bernoulli classification for the Furstenberg systems of $a(n)=e(G(n))$ and $b(n)=e(\alpha\lfloor G(n)\rfloor)$ and deriving zero-entropy topological corollaries along with an $L^2$ von Neumann-type convergence for commuting transformations. The approach circumvents traditional van der Corput methods due to the intermediate growth rate and hinges on a delicate derivative analysis of linear combinations of $G$, yielding robust equidistribution on $\mathbb{T}^r$ and $X^r$. Collectively, the results illuminate the probabilistic structure of intermediate-growth sequences, broaden the applicability of Furstenberg-system modeling, and connect ergodic properties to questions in number theory and topological dynamics.

Abstract

We give examples of sequences defined by smooth functions of intermediate growth, and we study the Furstenberg systems that model their statistical behavior. In particular, we show that the systems are Bernoulli. We do so by studying exponential sums that reflect the strong equidistribution properties of said sequences. As a by-product of our approach, we also get some convergence results.

Theorems & Definitions (50)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark
• Conjecture 1.6
• Conjecture 1.7
• Definition 2.1
• Remark