The translations of linear sets of fractional dimensions

Extract

If E is a linear set with a positive Lebesgue measure, and E(δ) is its translation through a distance δ, it is obvious that there always exist non-zero values of δ for which E. E(δ) has a positive measure. In the present note I show that for α-dimensional linear sets (0 < α < 1), and α-dimensional measure, the analogous result is not true. This is a further illustration of the deep structural difference between α-dimensional sets for α < 1 and for α = 1.