Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2015; 30(1): 7-21

Printed January 31, 2015

https://doi.org/10.4134/CKMS.2015.30.1.7

Copyright © The Korean Mathematical Society.

Singularity order of the Riesz-N{\'a}gy-Tak{\'a}cs function

In-Soo Baek

Busan University of Foreign Studies

Abstract

We give the characterization of H\"older differentiability points and non-differentiability points of the Riesz-N{\'a}gy-Tak{\'a}cs (RNT) singular function $\Psi_{a,p}$ satisfying $\Psi_{a,p}(a)=p$. It generalizes recent multifractal and metric number theoretical results associated with the RNT function. Besides, we classify the singular functions using the singularity order deduced from the H\"older derivative giving the information that a strictly increasing smooth function having a positive derivative Lebesgue almost everywhere has the singularity order 1 and the RNT function $\Psi_{a,p}$ has the singularity order $g(a,p)= \frac{a\log p +(1-a)\log (1-p)}{a\log a+(1-a) \log (1-a)}\geq 1$.

Keywords: Hausdorff dimension, packing dimension, distribution set, local dimension set, singular function, metric number theory, H\"older derivative

MSC numbers: Primary 28A78; Secondary 26A30