Projects per year
Abstract
We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R2 with Hausdorff dimension dimHK >1 such that the rotational components of the underlying similarities generate the full rotation group. Then, for all ε >0, writing πθ for projection onto the Lθ in direction θ, the Hausdorff dimensions of the sections satisfy dimH (K ∩ πθ-1x)> dimHK - 1 - ε for a set of x ∈ Lθ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.
Original language | English |
---|---|
Pages (from-to) | 13260-13289 |
Number of pages | 30 |
Journal | International Mathematics Research Notices |
Volume | 2015 |
Issue number | 24 |
Early online date | 15 Apr 2015 |
DOIs | |
Publication status | Published - 2015 |
Fingerprint
Dive into the research topics of 'Dimension conservation for self-similar sets and fractal percolation'. Together they form a unique fingerprint.Projects
- 1 Finished
-
Newton International Fellowship 2010: Newton International Fellowship 2010
Falconer, K. J. (PI)
1/01/11 → 31/12/12
Project: Fellowship