Continued fraction Cantor sets, Hausdorff dimension, and functional analysis (Q1183276)

Let \(A\) be a finite set of at least two positive integers. Then \(E(A)\) is the set of all irrational numbers with continued fraction expansion \(x=[a_1,a_2,\ldots]\) and \(a_j\in A\), \(j=1,2,3,\ldots\). The following generalization of the transfer operator related to continued fractions is considered: \[ L(s,A)f(t)=\sum_{k\in A}f\left(\frac{1}{k+t}\right)(k+t)^{-s}. \] Then there is a unique value \(s=s(A)\), \(0<s(A)<2\), such that \(L(s,A)^n1\) does not tend to 0 or \(\infty\) as \(n\to\infty\). Then \(\dim E(A)=2s(A)\). Therefore a careful analysis of the operator \(L(s,A)\) which is real hard work should give some estimate for \(\dim E(A)\). The main result of the paper states that for \(A_n=\{1,2,\ldots,n\}\) the asymptotic estimate \[ \dim E(A_ n)=1- \left(\frac{6}{\pi^2n}\right) - \left(\frac{72\log n}{\pi^ 4n^ 2}\right)+O\left(\frac{1}{n^2}\right) \] is valid. As indicated before the proof is hard analysis along the lines of previous work of Babenko, Mayer and Roepstorff and the author.