Metrics on N and the distribution of sequences
Milan Pa\v{s}t\'eka
TL;DR
This paper introduces a natural metric on natural numbers, explores conditions for uniform distribution of sequences, and characterizes continuity of distribution functions within this metric framework.
Contribution
It defines a natural metric on N, establishes conditions for uniform distribution, and characterizes sequence continuity, advancing understanding of metric-based sequence analysis.
Findings
A natural metric on N makes its completion a compact space.
Conditions for uniform distribution of sequences under this metric are established.
Continuity of distribution functions is characterized within this metric context.
Abstract
In the first part of this paper the notion of natural metric on the set of natural numbers is defined. It is such metric that the completion of N is a compact metric space that a probability borel measure exists in order that the sequence {n} is uniformly distributed. A necessary and sufficient condition where a given metric is natural. Later we study the properties of sequences uniformly continuous with respect to given natural metric. Inter alia Theorems 5 and 8 characterise the continuity of distribution function
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Stochastic processes and financial applications