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On the Hausdorff dimension of fibres

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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.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Itai Bejamini & Yuval Peres

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  1. Itai Bejamini
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  2. Yuval Peres
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Partially supported by a grant from the Landau Centre for Mathematical Analysis.

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Bejamini, I., Peres, Y. On the Hausdorff dimension of fibres. Israel J. Math. 74, 267–279 (1991). https://doi.org/10.1007/BF02775791

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  • DOI: https://doi.org/10.1007/BF02775791

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