On the Hausdorff dimension of fibres |
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Authors: | Itai Bejamini Yuval Peres |
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Institution: | (1) Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel |
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Abstract: | It is well known that there are planar sets of Hausdorff dimension greater than 1 which are graphs of functions, i.e., all
their vertical fibres consist of 1 point. We show this phenomenon does not occur for sets constructed in a certain “regular”
fashion. Specifically, we consider sets obtained by partitioning a square into 4 subsquares, discarding 1 of them and repeating
this on each of the 3 remaining squares, etc.; then almost all vertical fibres of a set so obtained have Hausdorff dimension
at least 1/2. Sharp bounds on the dimensions of sets of exceptional fibres are presented.
Partially supported by a grant from the Landau Centre for Mathematical Analysis. |
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Keywords: | |
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