[Paper Review] Badly Approximable Numbers and the Growth Rate of the Inclusion Length of an Almost Periodic Function

Mikhail Anikushin|arXiv (Cornell University)|Oct 9, 2017

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper establishes a lower bound for the growth rate of the inclusion length of quasiperiodic almost periodic functions using techniques from dimension theory, linking it to the Diophantine approximation properties of Fourier exponents. It shows that for badly approximable exponents, this lower bound coincides asymptotically with an upper bound derived from simultaneous Diophantine approximation, suggesting tightness of the estimate.

ABSTRACT

We study the growth rate of the inclusion length of an almost periodic function. For a given a. p. function such growth rate depends on the algebraic structure of Fourier exponents, i. e. on how good they can be approximated by rational numbers. In additional, as appears from the definition, the inclusion length carries some information about the translation numbers (almost periods). Our result is a lower bound of the growth rate of the inclusion interval of a quasiperiodic function (theorem 3). Here we use methods from dimension theory. We do not assume anything about exponents, but rationally independence. This suggest an idea that this lower bound can be reached (in asymptotic sense) for some "bad" exponents. Koichiro Naito in his papers on estimates of the fractal dimension of almost periodic attractors proved an upper bound of the inclusion length for some class of a.p. functions, using simultaneous Diophantine approximations. For the special case of badly approximable exponents we can see that the both estimates (if we consider them as asymptotic estimates) are coincide (see theorem 4). We hope that ideas and results presented in this paper can be useful not only to understand the nature of badly approximable numbers and almost periods, but also for more detailed understanding of the structure of almost periodic attractors.

Motivation & Objective

• To analyze the growth rate of the inclusion length of almost periodic functions based on the approximation properties of their Fourier exponents.
• To connect the inclusion length with translation numbers (almost periods) through algebraic structure of exponents.
• To establish a lower bound for the inclusion length growth rate under minimal assumptions—only rational independence of exponents.
• To investigate whether this lower bound is asymptotically sharp for 'badly approximable' exponents.
• To bridge results from dynamical systems and number theory, particularly in the context of almost periodic attractors.

Proposed method

• Uses methods from dimension theory to analyze the inclusion length of almost periodic functions.
• Applies tools from Diophantine approximation, particularly focusing on badly approximable numbers.
• Considers the algebraic structure of Fourier exponents and their rational approximation quality.
• Derives a lower bound for the inclusion length growth rate without assuming any specific arithmetic structure beyond rational independence.
• Compares the derived lower bound with an upper bound from prior work by Koichiro Naito on fractal dimension of almost periodic attractors.
• Employs asymptotic analysis to compare the lower and upper bounds in the special case of badly approximable exponents.

Experimental results

Research questions

• RQ1 How does the growth rate of the inclusion length of an almost periodic function depend on the Diophantine properties of its Fourier exponents?
• RQ2 Can a non-trivial lower bound be established for the inclusion length growth rate under only rational independence of exponents?
• RQ3 Is the derived lower bound asymptotically tight for the class of badly approximable exponents?
• RQ4 To what extent do the inclusion length and translation numbers (almost periods) reflect the arithmetic structure of the Fourier spectrum?
• RQ5 Can the interplay between number theory and dynamical systems be leveraged to better understand the structure of almost periodic attractors?

Key findings

• A lower bound for the growth rate of the inclusion length of a quasiperiodic function is established under the sole assumption of rational independence of Fourier exponents.
• The lower bound is shown to be asymptotically sharp for the class of badly approximable exponents, matching an upper bound from prior work on fractal dimension of attractors.
• The inclusion length growth rate is fundamentally tied to the quality of rational approximation of Fourier exponents, particularly in the badly approximable case.
• The results suggest that badly approximable exponents represent a critical case where the inclusion length growth rate achieves a minimal possible asymptotic rate.
• The connection between inclusion length and translation numbers is confirmed through the algebraic structure of the exponents.
• The findings support the idea that number-theoretic properties of exponents govern the geometric and dynamical complexity of almost periodic functions.