Projecting the one-dimensional Sierpiński gasket
type: Item
instance of: scholarly article
title: Projecting the one-dimensional Sierpiński gasket
zbMATH Open document ID: 0871.28006
DOI: 10.1007/BF02774038
published in: Israel Journal of Mathematics
publication date: 1997-08-31T00:00:00Z
review text: The paper under review deals with the projections of the Sierpiński gasket. One considers the linear projection \(\pi_u\) of \({\mathbb{R}}^2\) onto the \(x\)-axis that maps \((0,1)\) to the point \((u,0)\). Denote \(S_u=\pi_u(S)\). The author computes the Lebesgue measure and the Hausdorff dimension of the projection \(S_u\). He proves that the projection of \(S\) in any irrational direction has Lebesgue measure equal to zero, while the projection in a rational direction \(p/q\) has Hausdorff dimension less than 1 unless \(p+q=0 \text{mod} 3\), in which case the projection has Lebesgue measure equal to \(1/q\). A lower bound for the Hausdorff dimension of \(S_u\) is also given in terms of the approximants of \(u\) by rationals. Moreover, it is constructed a residual set of directions on which the Hausdorff dimension of \(S_u\) is 1.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 1011400
zbMATH Keywords: projection of the Sierpiński gasket
zbMATH Keywords: Lebesgue measure
zbMATH Keywords: Hausdorff dimension
author: Richard Kenyon
reviewed by: Emilia Petrisor
MaRDI profile type: Publication
cites work: A Note on Minimal Covers for Sofic Systems
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description: scientific article; zbMATH DE number 1011400