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Abstract
We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic selfsimilar sets. For example, let K be a selfsimilar subset of R^{2} with Hausdorff dimension dim_{H}K >1 such that the rotational components of the underlying similarities generate the full rotation group. Then, for all ε >0, writing π_{θ} for projection onto the L_{θ} in direction θ, the Hausdorff dimensions of the sections satisfy dim_{H} (K ∩ π_{θ}^{1}x)> dim_{H}K  1  ε for a set of x ∈ L_{θ} of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of selfsimilar 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 (fromto)  1326013289 
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 
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Dive into the research topics of 'Dimension conservation for selfsimilar sets and fractal percolation'. Together they form a unique fingerprint.Projects
 1 Finished

Newton International Fellowship 2010: Newton International Fellowship 2010
1/01/11 → 31/12/12
Project: Fellowship