TL;DR
This paper investigates harmonic ultrafilters, which are ultrafilters composed of harmonic sets of natural numbers, demonstrating their structure as a compact semigroup and identifying its smallest ideal, extending Hindman's work.
Contribution
It extends Hindman's research by characterizing harmonic ultrafilters as a compact semigroup and determining its smallest ideal.
Findings
Harmonic ultrafilters form a compact semigroup under Glazer addition.
The smallest ideal of this semigroup is identified.
The work builds on and extends Hindman's foundational results.
Abstract
A set of natural numbers will be called \emph{harmonic} if the reciprocals of its elements form a divergent series. An ultrafilter of the natural numbers will he called \emph{harmonic} if all each members are harmonic sets. The harmonic ultrafilters are shown to constitute a compact semigroup under the Glazer addition and its smallest ideal is obtained. This paper is an extension of work begun by Hindman. Alle ingredients are found in the treatise by Hindman and Strauss.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Limits and Structures in Graph Theory · Analytic Number Theory Research