Slicing Sets and Measures, and the Dimension of Exceptional Parameters

We consider the problem of slicing a compact metric space Ωwith sets of the form π_λ^{-1}\{t\}, where the mappings π_λ \colon Ω\to \R, λ\in \R, are \emph{generalized projections}, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that Ωhas Hausdorff dimension strictly greater than one, what is the dimension of the 'typical' slice π_λ^{-1}{t}, as the parameters λand t vary. In the special case of the mappings π_λ being orthogonal projections restricted to a compact set Ω\subset \R^{2}, the problem dates back to a 1954 paper by Marstrand: he proved that for almost every λthere exist positively many $t \in \R$ such that \dim π_λ^{-1}{t} = \dim Ω- 1. For generalized projections, the same result was obtained 50 years later by Järvenpää, Järvenpää and Niemelä. In this paper, we improve the previously existing estimates by replacing the phrase 'almost all λ' with a sharp bound for the dimension of the exceptional parameters.

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