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The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. (English) Zbl 0811.28005
Summary: We show that the self-affine sets considered by C. McMullen [Nagoya Math. J. 96, 1-9 (1984; Zbl 0539.28003)] and by T. Bedford [Ph. D. Thesis (1984)] have infinite Hausdorff measure in their dimension, except in the (rare) cases where the Hausdorff dimension coincides with the Minkowski (\(\equiv\) box) dimension. More precisely, the Hausdorff measure of such a self-affine set \(K\) is infinite in the gauge \[ \phi(t)= t^ \gamma\exp\left[{- c|\log t|\over (\log| \log t|)^ 2}\right], \] (where \(\gamma\) is the Hausdorff dimension of \(K\), and \(c> 0\) is small). The Hausdorff measure of \(K\) becomes zero if 2 is replaced by any smaller number in the formula for the gauge \(\phi\).

MSC:
28A80 Fractals
28A78 Hausdorff and packing measures
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