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 graphs.
We say that \(G \subset {\mathbb{R}^k} \times {\mathbb{R}^n}\) is Γk-null if for every Lipschitz function \(f:{\mathbb{R}^k} \to {\mathbb{R}^n}\) the set \(\{t \in {\mathbb{R}^k}:(t,f(t)) \in G\} \) has measure zero. We show that for every Borel set \(E \subset {\mathbb{R}^k} \times {\mathbb{R}^n}\) with dim \(({\rm{pro}}{{\rm{j}}_{{\mathbb{R}^k}}}E) = k\) there is a Γk-null subset G ⊂ E such that
where ess- sup(dim Et) is the essential supremum of the Hausdorff dimension of the vertical sections \({\{{E_t}\} _{t \in {\mathbb{R}^k}}}\) of E.
In addition, we show that, provided that E is not Γk-null, there is a Γk-null subset G ⊂ E such that for F = E G, the Fubini property holds, that is, dim (F) = k + ess-sup(dim Ft).
We also obtain more general results by replacing ℝk by an Ahlfors–David regular set. Applications of our results include Fubini-type results for unions of affine subspaces, connection to the Kakeya conjecture and projection theorems.
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This research was supported by the Hungarian National Research, Development and Innovation Office – NKFIH, 124749.
The second author is grateful to the Alfréd Rényi Institute, where he was a visiting researcher during a part of this project. He was also partially supported by the Hungarian National Research, Development and Innovation Office – NKFIH, 129335.
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Héra, K., Keleti, T. & Máthé, A. A Fubini-type theorem for Hausdorff dimension. JAMA 152, 471–506 (2024). https://doi.org/10.1007/s11854-023-0302-3
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DOI: https://doi.org/10.1007/s11854-023-0302-3