Dimension Theory Approach to the Complexity of Almost Periodic Trajectories

TL;DR

This paper introduces the Diophantine dimension as a new measure of complexity for almost periodic functions, linking it to recurrence, ergodicity, and quasiperiodic behavior, with applications to evolution equations.

Contribution

It defines the Diophantine dimension and explores its properties, estimates, and connections to classical theorems, providing new tools for analyzing almost periodic trajectories.

Findings

Estimates of Diophantine dimension for quasiperiodic functions

Methods to analyze almost periodic trajectories of evolution equations

Connections to effective Kronecker theorem

Abstract

We introduce and study a dimensional-like characteristic of an uniformly almost periodic function, which we call the Diophantine dimension. By definition, it is the exponent in the asymptotic behavior of the inclusio length. Diophantine dimension is connected with recurrent and ergodic properties of an almost periodic function. We get some estimates of the Diophantine dimension for certain quasiperiodic functions and present methods to investigate such a characteristic for almost periodic trajectories of evolution equations. Also we discuss the link between the presented approach and the so called effective versions of the Kronecker theorem.