Affine embeddings of Cantor sets and dimension of αβ-sets

2016/09/19 by Feng, De-Jun, Xiong, Ying
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.05784

Abstract

Let E, F⊂ \Bbb R^d be two self-similar sets, and suppose that F can be affinely embedded into E. Under the assumption that E is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of E and F. This gives a partial affirmative answer to Conjecture 1.2 in \citeFHR14. The proof is based on our study of the box-counting dimension of a class of multi-rotation invariant sets on the unit circle, including the αβ-sets initially studied by Engelking and Katznelson.

Related

• Affine embeddings and intersections of Cantor sets
• Affine embeddings of Cantor sets on the line
• Affine embeddings of Cantor sets in the plane
• New dimension bounds for αβ sets
• Resonance between Cantor sets