Riesz capacities of a set due to Dobiński

Abstract

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

Keywords

• Riesz capacity
• Logarithmic capacity
• Dobiński set
• Dyadic capacity
• Non-linear capacity
• Diophantine approxmation