On restricted families of projections in R^3

Abstract

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 $\epsilon$, 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 ${\dim }_{\mathrm{H}}K\le 1$ 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.

33 pages. v2: small changes, including extended introduction and additional references. To appear in Proc. London Math. Soc

References in corpus (3)

• On self-similar sets with overlaps and inverse theorems for entropy
• Application of a Fourier restriction theorem to certain families of projections in R^3
• Hausdorff dimension estimates for restricted families of projections in ${\mathbb{R}}^3$

Cited by in corpus (26)

• Application of a Fourier restriction theorem to certain families of projections in R^3
• On the Assouad dimension of projections
• The Assouad dimensions of projections of planar sets
• A Marstrand-type restricted projection theorem in ${\mathbb{R}}^3$
• Projections of planar sets in well-separated directions
• The Assouad dimension of Kakeya sets in ${\mathbb{R}}^3$
• Combinatorial proofs of two theorems of Lutz and Stull
• Improved bounds for restricted projection families via weighted Fourier restriction
• Orthogonal projections of discretized sets
• Improved bounds for restricted families of projections to planes in ${\mathbb{R}}^3$
• Improving Kaufman's exceptional set estimate for packing dimension
• Hausdorff dimension estimates for restricted families of projections in ${\mathbb{R}}^3$
• Restricted families of projections onto planes: The general case of nonvanishing geodesic curvature
• Dimension Distortion by Right Coset Projections in the Heisenberg Group
• A discretised projection theorem in the plane
• Sticky Kakeya sets and the sticky Kakeya conjecture
• Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem
• Restricted families of projections in vector spaces over finite fields
• A combinatorial proof of a sumset conjecture of Furstenberg
• Projection theorems in hyperbolic space
• Marstrand-type projection theorems for linear projections and in normed spaces
• Length of sets under restricted families of projections onto lines
• A Fubini-type theorem for Hausdorff dimension
• Distance sets, orthogonal projections, and passing to weak tangents
• Cantor series expansions and packing dimension faithfulness
• The dimension of projections induced by a curve