On restricted families of projections in ℝ3

By Katrin Fässler, Tuomas Orponen. 2013

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We study projections onto non‐degenerate one‐dimensional families of lines and planes in R3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 ‐dimensional sets B⊂R3 is typically preserved under one‐dimensional families of projections onto lines. We improve the result by an ε , proving that if dimHB=s>12 , then the packing dimension of the projections is almost surely at least σ(s)>12 . For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self‐similar sets K⊂R3 without rotations, we obtain a full Marstrand‐type projection theorem for 1‐parameter families of projections onto lines. The dimHK⩽1 case of the result follows from recent work of M. Hochman, but the dimHK>1 part is new: with this assumption, we prove that the projections have positive length almost surely.

Subfields: Computer Vision and Pattern Recognition, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9fb2b721e43fefcc47e2f14c48407663e8ce4f57