Chain logarithm

A chain logarithm is an object of real analysis that, like a continued fraction , allows a representation of real numbers. A finite chain logarithm to an integer base is an expression of the form ${\displaystyle m\geq 3}$

${\displaystyle K\log _{m}(d_{1},\dots ,d_{n}):=\log _{m}(d_{1}+\log _{m}(d_{2}+\log _{m}(\dots +\log _{m}(d_{n})\dots ),}$

with for . denotes the logarithm to the base of . An infinite chain logarithm is through the limit ${\displaystyle d_{k}\in \{1,\dots ,m-1\}}$ ${\displaystyle k=1,\dots n}$ ${\displaystyle \log _{m}(x)}$ ${\displaystyle m}$ ${\displaystyle x}$

${\displaystyle \lim _{n\to \infty }{\it {K\log }}_{m}(d_{1},\dots ,d_{n}),}$

for an episode in , given. For a basis every real number in has a representation by an infinite chain logarithm and this representation is unique except for a countable set of real numbers. In the representation of almost all real numbers in , in relation to the Lebesgue measure , all numbers appear in infinitely often. On the other hand, almost all numbers are abnormal in terms of being represented as chain logarithms, i.e. H. the numbers in do not occur with the same frequency. ${\displaystyle \{1,\dots ,m-1\}^{\mathbb {N} }}$ ${\displaystyle m\geq 3}$ ${\displaystyle [0,1]}$ ${\displaystyle [0,1]}$ ${\displaystyle \{1,\dots ,m-1\}}$ ${\displaystyle \{1,\dots ,m-1\}}$

Individual evidence

1. ^ Jörg Neunhäuserer: Continued logarithm representation of real numbers. Real Analysis Exchange 42, 2018, pp. 57-66.