Projecting the one-dimensional Sierpinski gasket

Let S subset of R-2 be the Canter 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 of S in any irrational direction has Lebesgue measure 0. The projection in a rational direction pig has Hausdorff dimension less than 1 unless p + 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.