Intersections of middle- α Cantor sets with a fixed translation.

For λ ∈ (0 , 1 / 3 ] let C λ be the middle- (1 − 2 λ) Cantor set in R . Given t ∈ [ − 1 , 1 ] , excluding the trivial case we show that Λ (t) := λ ∈ (0 , 1 / 3 ] : C λ ∩ (C λ + t) ≠ ∅ is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of Λ (t) , which reveals a dimensional variation principle. Furthermore, for any β ∈ [ 0 , 1 ] we show that the level set Λ β (t) : = { λ ∈ Λ (t) : dim H (C λ ∩ (C λ + t)) = dim P (C λ ∩ (C λ + t)) = β log 2 − log λ } has equal Hausdorff and packing dimension (− β log β − (1 − β) log 1 − β 2) / log 3. We also show that the set of λ ∈ Λ (t) for which dim H (C λ ∩ (C λ + t)) ≠ dim P (C λ ∩ (C λ + t)) has full Hausdorff dimension. [ABSTRACT FROM AUTHOR]