The monoid of numbers of the form 1 < a^q /b^p < a

2023/12/11 by Laurent Fallot, Fallot, Laurent
Computer Science · Mathematics · #Analytic Number Theory Research #FOS: Mathematics #General Topology (math.GN) #History and Theory of Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2312.06248

openalex publication_date 2023/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is a study of the set of rational numbers of the form 1 &lt; a^q /b^p &lt; a with a and b co-prime integers. The set F (a,b) of these numbers, with an appropriate binary law, is a monoid isomorphic to (N, +, 0). We identify the sequences of minimum and maximum record holders in F (a,b) and prove that the first one converges to 1 while the second one converges to a. We conclude that F (a,b) is dense in the set of the real numbers comprise between 1 and a.

Related

• Factorizations in reciprocal Puiseux monoids
• Characterisation of (Sub)sequential Rational Functions over a General Class Monoids
• On a problem of Romanoff type
• On the atomicity of power monoids of Puiseux monoids
• Framed Blob Monoids