On the Assouad dimension of projections

Abstract

Let $F\subset {\mathbb{R}}^2$, and let ${\dim }_{\mathrm{A}}$ stand for Assouad dimension. I prove that ${\dim }_{\mathrm{A}}{\pi }_e(F)\ge \min \{{\dim }_{\mathrm{A}}F,1\}$ for all $e\in S^1$ outside of a set of Hausdorff dimension zero. This is a strong variant of Marstrand's projection theorem for Assouad dimension, whose analogue is not true for other common notions of fractal dimension, such as Hausdorff or packing dimension.

37 pages, 3 figures. v3: small corrections

References in corpus (3)

• Rigidity of quasisymmetric mappings on self-affine carpets
• Projections of Gibbs measures on self-conformal sets
• Attainable values for the Assouad dimension of projections

Cited by in corpus (4)

• Finer geometry of planar self-affine sets
• Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line
• Attainable values for the Assouad dimension of projections
• Assouad dimension and local structure of self-similar sets with overlaps in ${\mathbb{R}}^d$