[Paper Review] Almost periodic discrete sets

S. Favorov, Ye. Kolbasina|arXiv (Cornell University)|Jan 30, 2010

advanced mathematical theories4 citations

TL;DR

This paper introduces a geometric and metric-based framework for characterizing almost periodic discrete sets in ℝᵏ using a specialized distance metric between sequences. It establishes the completeness of the space of almost periodic sets and proves an analogue of the Bochner criterion for almost periodicity, showing that a discrete set is almost periodic if and only if its associated Dirac measure is almost periodic, thereby linking discrete sets to almost periodic measures.

ABSTRACT

Using a special metric in the space of sequences, we give a geometric description of almost periodic sets in the $k$-dimensional Euclidean space. We prove the completeness of the space of almost periodic sets and some analogue of the Bochner criterion of almost periodicity. Also, we show the connection between these sets and almost periodic measures.

Motivation & Objective

• To provide a geometric and metric-based characterization of almost periodic discrete sets in k-dimensional Euclidean space.
• To establish the completeness of the space of almost periodic sets under a newly defined metric on sequences.
• To develop an analogue of the Bochner criterion for almost periodicity in the context of discrete sets.
• To clarify the connection between almost periodic discrete sets and almost periodic measures.

Proposed method

• Defining a distance between discrete multiple sets using the infimum over all bijections of the supremum of pairwise point distances.
• Introducing ε-almost periods for discrete sets as translations that preserve proximity under some bijection.
• Using weak uniform convergence of measures and convolution with compactly supported continuous functions to analyze limit behavior.
• Applying the Bochner criterion for almost periodic functions to sequences of translated sets and their associated measures.
• Proving convergence of measures via uniform approximation using mollifiers (e.g., φ*μ) and compact support arguments.
• Establishing equivalence between almost periodicity of a discrete set and that of its associated Dirac measure.

Experimental results

Research questions

• RQ1 What geometric and metric conditions characterize almost periodic discrete sets in ℝᵏ?
• RQ2 Is the space of almost periodic discrete sets complete under the proposed metric?
• RQ3 Can a Bochner-type criterion be formulated for almost periodic discrete sets?
• RQ4 How are almost periodic discrete sets related to almost periodic measures?
• RQ5 Under what conditions does convergence of discrete sets imply convergence of their associated measures?

Key findings

• The space of almost periodic discrete sets is complete under the defined metric on sequences.
• A discrete set is almost periodic if and only if its associated Dirac measure is almost periodic.
• An analogue of the Bochner criterion holds: a discrete set is almost periodic iff every sequence of translations has a subsequence such that the translated sets converge in the defined metric.
• Weak uniform convergence of measures associated with translated sets implies convergence of the sets themselves, under the metric.
• For any compactly supported continuous function φ, the convolution φ*μ_D converges uniformly as the set is translated, under convergence of the sets.
• The number of points of a discrete set within any bounded set is uniformly bounded under convergence, ensuring topological stability.