On the structure of the intersection of two middle third Cantor sets (Q1903616)

Main results: 1) Let \(C\) be the (ordinary) Cantor set in \([0,1]\). Then for almost all \(x\) in \([0, 1]\) (with respect to Lebesgue's measure \(m\) in \([0,1]\) the set \(({x}+ C)\cap C\) is uncountable. 2) For every \(\alpha\) such that \(0\leq \alpha\leq 1\), there exists an \(x\) in \([0, 1]\) such that the Hausdorff dimension of \(({x}+C)\cap C\) satisfies \[ \dim(({x}+ C)\cap C)= (1- \alpha)\ln 2/\ln 3. \] Moreover, the set \(D_\alpha\) of such \({x}\)'s is a dense subset of \([0, 1]\), and either \({m}(D_\alpha)= 0\) or \({m}(D_\alpha)= 1\).