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Projecting the one-dimensional Sierpinski gasket

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Abstract

LetS⊂ℝ2 be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1.

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References

  1. A. S. Besicovich,On the fundamental geometric properties of linearly measurable plane sets of points III, Mathematische Annalen116 (1939), 349–357.

    Article  MathSciNet  Google Scholar 

  2. M. Boyle, B. Kitchens and B. Marcus,A note on minimal covers for sofic systems, Proceedings of the American Mathematical Society95 (1985), 403–411.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. J. Falconer,The Geometry of Fractal Sets, Cambridge Tracts in Mathematics85, 1985.

  4. K. J. Falconer,Dimensions and measures of quasi self-similar sets, Proceedings of the American Mathematical Society106 (1989), 543–554.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Kenyon,Self-replicating tilings, inSymbolic Dynamics and its Applications, Contemporary Mathematics series, Vol. 135 (P. Walters, ed.), American Mathematical Society, Providence, RI, 1992.

    Google Scholar 

  6. M. Keane and M. Smorodinski,β-expansions, preprint.

  7. J. Lagarias and Y. Wang,Tiling the line with one tile, Preprint, ATT Bell Labs.

  8. J. Marstrand,Some fundamental geometric properties of plane sets of fractional dimension, Proceedings of the London Mathematical Society4 (1954), 257–302.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Odlyzko,Nonnegative digit sets in positional number systems, Proceedings of the London Mathematical Society37 (1978), 213–229.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CNRS UMR 128, Ecole Normale Superieure de Lyon, 46, allée d’Italie, 69364, Lyon, France

    Richard Kenyon

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  1. Richard Kenyon
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Correspondence to Richard Kenyon.

Additional information

This work was partially completed while the author was at the Institut Fourier, Grenoble, France.

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Kenyon, R. Projecting the one-dimensional Sierpinski gasket. Isr. J. Math. 97, 221–238 (1997). https://doi.org/10.1007/BF02774038

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  • DOI: https://doi.org/10.1007/BF02774038

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