Capacities of certain Cantor sets
The author's main results are a necessary and also sufficient condition for a generalized Cantor set [cf. \textit{M. Ohtsuka}, Nagoya Math. J. 11, 151-160 (1957; Zbl 0091.27501)] to have positive capacity. Next, he establishes, by means of examples, that there exist Cantor-type sets \(K\) with 0 capacity, while the projection of the square set \(K\times K\) in all but two directions have positive capacity. A direction \(e^{i\theta}\) \((\theta\in [0, 2\pi])\) of the unit circle will be called dyadic if it can be represented in the form \(\theta =\displaystyle\sum ^m_{k=0}\displaystyle{{a_k}\over{2^h}}, m=m(\theta)\), \(a_k\in\{0,1\}\), \(a_m=1\) if \(\theta\not= 0\). Finally, the author shows there exists a Cantor set of positive capacity such that the capacity \(\text{cap} P_{\theta}(K)=0\) for every projection \(P_{\theta}\) in a dyadic direction \(e^{i\theta}\). See also the review Zbl 0880.31002 below.
scientific article; zbMATH DE number 1052952