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

On restricted families of projections in R^3

We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most $1/2$-dimensional sets $B \subset \mathbb{R}^{3}$ is typically preserved under one-dimensional families of projections onto lines. We improve the result by an $\varepsilon$, proving that if $\dim_{\mathrm{H}} B = s > 1/2$, then the packing dimension of the projections is almost surely at least $\sigma(s) > 1/2$. For projections onto planes, we obtain a similar bound, with the threshold $1/2$ replaced by $1$. In the special case of self-similar sets $K \subset \mathbb{R}^{3}$ without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The $\dim_{\mathrm{H}} K \leq 1$ case of the result follows from recent work of M. Hochman, but the $\dim_{\mathrm{H}} K > 1$ part is new: with this assumption, we prove that the projections have positive length almost surely.Comment: 33 pages. v2: small changes, including extended introduction and additional references. To appear in Proc. London Math. So