, Volume 97, Issue 1, pp 221-238

Projecting the one-dimensional Sierpinski gasket

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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.

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