Slicing Sets and Measures, and the Dimension of Exceptional Parameters

TL;DR

This paper investigates the dimension of slices of a compact metric space under generalized projections, providing sharper bounds on the size of exceptional parameter sets where typical dimension results do not hold.

Contribution

It improves existing estimates by establishing a precise bound on the dimension of the set of exceptional parameters for generalized projections.

Findings

Established a sharp bound for the dimension of exceptional parameters.

Extended classical results to generalized projections beyond orthogonal projections.

Enhanced understanding of the structure of slices in metric spaces.

Abstract

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 $t \in \R$ 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.