Intersections of middle-α Cantor sets with a fixed translation

Abstract

For let C λ be the middle- Cantor set in . Given , excluding the trivial case we show that Λ(t):=λ∈(0,1/3]:Cλ∩(Cλ+t)≠∅ is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of , which reveals a dimensional variation principle. Furthermore, for any we show that the level set Λβ(t):={λ∈Λ(t):dimH(Cλ∩(Cλ+t))=dimP(Cλ∩(Cλ+t))=βlog2−logλ} has equal Hausdorff and packing dimension . We also show that the set of for which has full Hausdorff dimension.