Bulletin of the
Korean Mathematical Society BKMS

We give characterizations of the differentiability points and the non-differentiability points of the Riesz-N{\'a}gy-Tak{\'a}cs(RNT) singular function using the distribution sets in the unit interval. Using characterizations, we show that the Hausdorff dimension of the non-differentiability points of the RNT singular function is greater than $0$ and the packing dimension of the infinite derivative points of the RNT singular function is less than $1$. Further the RNT singular function is nowhere differentiable in the sense of topological magnitude, which leads to that the packing dimension of the non-differentiability points of the RNT singular function is 1. Finally we show that our characterizations generalize a recent result from the $(\tau, \tau -1)$-expansion associated with the RNT singular function adding a new result for a sufficient condition for the non-differentiability points.

Keywords: Hausdorff dimension, packing dimension, distribution set, local dimension set, singular function, metric number theory

MSC numbers: 26A30, 28A80