Riesz capacities of a set due to Dobi\'nski

TL;DR

This paper investigates the Riesz capacity of the Dobiński set, characterizing parameter conditions for positivity and establishing its positive logarithmic capacity, thereby answering a previously open question.

Contribution

It provides a characterization of parameter ranges for positive Riesz capacity of the Dobiński set and proves its positive logarithmic capacity, using dyadic analogues.

Findings

Dobiński set has positive logarithmic capacity.

Characterization of parameters for positive Riesz capacity.

Dyadic analogues are effective in the analysis.

Abstract

We study the Riesz $(a,p)$-capacity of the so called Dobi\'nski set. We characterize the values of the parameters $a$ and $p$ for which the $(a,p)$-Riesz capacity of the Dobi\'nski set is positive. In particular we show that the Dobi\'nski set has positive logarithmic capacity, thus answering a question of Dayan, Fernand\'ez and Gonz\'alez. We approach the problem by considering the dyadic analogues of the Riesz $(a,p)$-capacities which seem to be better adapted to the problem.