aequationes mathematicae

, Volume 64, Issue 3, pp 232–247 | Cite as

On certain dense sets of rationals, simultaneous Schröder and Böttcher equations, and characterizations of functions

  • A. Sklar


The fact that rational numbers of the forms 2 -m 3 n , m and n integers, are dense in the set \( \mathbb{R}^+ \) of non-negative real numbers is crucial in determining well-behaved solutions of a key functional equation. A principal aim of this paper is the presentation of a new proof of the statement that many similar sets of rationals are dense in \( \mathbb{R}^+ \). The reason for giving a new proof of this statement is that the “standard” argument uses all the basic properties of logarithms and exponentials. The new proof does not, which means that our result can be used without circularity not only in the characterization, but in the very definition of logarithms and exponential functions.

Keywords. Dense sets of rationals, logarithms, exponential functions, Schröder equation, Böttcher equation. 


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Copyright information

© Birkhäuser Verlag, Basel, 2002

Authors and Affiliations

  • A. Sklar
    • 1
  1. 1.Department of Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA¶ private: 5044 Marine Drive, Chicago, Illinois 60616, USA US

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