On restricted families of projections in R3

We study projections onto non-degenerate one-dimensional families A lines and planes in R-3. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorfl dimension of at most 1/2-dimensional sets B subset of R-3 is typically preserved under one-dimensional families of projections onto lines. We improve the result by an e, proving that if dim(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 of R-3 without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The dimH K <= 1 case of the result follows from recent work of NI. Hochman, but the dimu K > 1 part is new: with this assumption, we prove that the projections have positive length almost surely.