A Fubini-type theorem for Hausdorff dimension

Héra, Kornélia and Keleti, Tamás and Máthé, András (2023) A Fubini-type theorem for Hausdorff dimension. JOURNAL D ANALYSE MATHEMATIQUE. ISSN 0021-7670

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Abstract

It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz curves/surfaces. We say that G⊂Rk×Rn is Γk-null if for every Lipschitz function f:Rk→Rn the set {t∈Rk:(t,f(t))∈G} has measure zero. We show that for every compact set E⊂Rk×Rn there is a Γk-null subset G⊂E such that dim(E∖G)=k+ess-sup(dimEt) where ess-sup(dimEt) is the essential supremum of the Hausdorff dimension of the vertical sections {Et}t∈Rk of E, assuming that projRkE has positive measure. We also obtain more general results by replacing Rk by an Ahlfors regular set. Applications of our results include Fubini-type results for unions of affine subspaces and related projection theorems.

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