Lacunary self-similar fractal sets and intersection of Cantor sets (Q1394503)

Let \(K\) be the Cantor set in \([0,1]\). In this paper the author studies mainly the Hausdorff dimension of the intersection of \(K\) and its translation \(K+a\), where \(a\in [0,1]\). To this end, the author introduces the notion of a lacunary self-similar set. Every self-similar set is lacunary self-similar. It is shown that for almost all \(a\in [0,1]\), the set \(K\cap (K+a)\) is lacunary self-similar and has Hausdorff dimension \(\ln 2 / 3\ln 3\). Finally, this result is generalized to sets of the form \(K^p\cap (K^p+a)\), where \(a\in [0,1]\) and \(K^p\) is the set of numbers in \([0,1]\) that admit base \(p\)-expansions without odd digits.