ABSTRACT. We study projections onto non-degenerate one-dimensional families of lines and planes in 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 ⊂ R 3 is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε, proving that if dim H B = s > 1/2, then the packing dimension of the projections is almost surely at least σ(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 ⊂ R 3 without rotations, we obtain a full Marstrand type projection theorem for oneparameter families of projections onto lines. The dim H K ≤ 1 case of the result follows from recent work of M. Hochman, but the dim H K > 1 part is new: with this assumption, we prove that the projections have positive length almost surely. CONTENTS

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces.To introduce the first one, assume that E, K ⊂ R 2 are non-empty Borel sets with dimH K > 0. Does the radial projection of K to some point in E have positive dimension? Not necessarily: E can be zero-dimensional, or E and K can lie on a common line. I prove that these are the only obstructions: if dimH E > 0, and E does not lie on a line, then there exists a point in x ∈ E such that the radial projection πx(K) has Hausdorff dimension at least (dimH K)/2. Applying the result with E = K gives the following corollary: if K ⊂ R 2 is Borel set, which does not lie on a line, then the set of directions spanned by K has Hausdorff dimension at least (dimH K)/2.For the second result, let d ≥ 2 and d − 1 < s < d. Let µ be a compactly supported Radon measure in R d with finite s-energy. I prove that the radial projections of µ are absolutely continuous with respect to H d−1 for every centre in R d \ spt µ, outside an exceptional set of dimension at most 2(d − 1) − s. In fact, for x outside an exceptional set as above, the proof shows that π x♯ µ ∈ L p (S d−1 ) for some p > 1. The dimension bound on the exceptional set is sharp.A Borel set K ⊂ R 2 will be called 2010 Mathematics Subject Classification. 28A80 (Primary) 28A78 (Secondary).

For 1 < p < ∞, weight w ∈ A p and any L 2 -bounded Calderón-Zygmund