The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

Iteration of the rational function z-1/z and a Hausdorff moment sequence

In a previous paper we considered a positive function f, uniquely determined for s>0 by the requirements f(1)=1, log(1/f) is convex and the functional equation f(s)=psi(f(s+1)) with psi(s)=s-1/s. We prove that the meromorphic extension of f to the whole complex plane is given by the formula f(z)=lim_{n\to\infty}psi^{\circ n}(lambda_n(lambda_{n+1}/lambda_n)^z), where the numbers lambda_n are defined by lambda_0=0 and the recursion lambda_{n+1}=(1/2)(lambda_n+sqrt{lambda_n^2+4}). The numbers m_n=1/lambda_{n+1} form a Hausdorff moment sequence of a probability measure μsuch that \int t^{z-1}dμ(t)…

Rational Extensions of C(X) via Hausdorff Continuous Functions

The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.

Hausdorff Measures, Dyadic Approximations and Dobiński Set

Dobiński set $\mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of $\mathcal{D}$ by means of the Mass Transference Principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in $\mathcal{D}$.

关于 Riemann 不可微函数的 Hausdorff 维数

On the Hausdorff dimension of Riemann's non-differentiable function

本文研究 Riemann 不可微函数的 Hausdorff 维数，通过重新计算其在有理点附近的渐近行为并联系 Talbot 效应与 Gauss 和，给出了维数上界估计并推广至多重分形设定。

Sets of large dimension not containing polynomial configurations

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set $E\subset \R^n$, we study the angles determined by three points of $E$. The main result implies the existence of a compact set in $\R^n$ of Hausdorff dimension $n/2$ which does not contain the angle $π/2$. (This is known to be sharp if $n$ is even.) We show that …