5 research outputs found
Slicing Sets and Measures, and the Dimension of Exceptional Parameters
We consider the problem of slicing a compact metric space \Omega with sets of
the form \pi_{\lambda}^{-1}\{t\}, where the mappings \pi_{\lambda} \colon
\Omega \to \R, \lambda \in \R, are \emph{generalized projections}, introduced
by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that
\Omega has Hausdorff dimension strictly greater than one, what is the dimension
of the 'typical' slice \pi_{\lambda}^{-1}{t}, as the parameters \lambda and t
vary. In the special case of the mappings \pi_{\lambda} being orthogonal
projections restricted to a compact set \Omega \subset \R^{2}, the problem
dates back to a 1954 paper by Marstrand: he proved that for almost every
\lambda there exist positively many such that \dim
\pi_{\lambda}^{-1}{t} = \dim \Omega - 1. For generalized projections, the same
result was obtained 50 years later by J\"arvenp\"a\"a, J\"arvenp\"a\"a and
Niemel\"a. In this paper, we improve the previously existing estimates by
replacing the phrase 'almost all \lambda' with a sharp bound for the dimension
of the exceptional parameters.Comment: 31 pages, three figures; several typos corrected and large parts of
the third section rewritten in v3; to appear in J. Geom. Ana
Self-similar sets: projections, sections and percolation
We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.Postprin