Non-existence of multi-line Besicovitch sets

If a compact set K \subset R^2 contains a positive-dimensional family of line-segments in positively many directions, then K has positive measure.Comment: 7 pages. v2: corrected typos and added references, to appear in Publ. Ma

Tangent measures of non-doubling measures

We construct a non-doubling measure on the real line, all tangent measures of which are equivalent to Lebesgue measure.Comment: 17 pages, 5 figures. v2: Minor corrections throughout, and section six completely rewritten in a more reader-friendly style; Accepted to Math. Proc. Cambridge Philos. So