2021/06/17 by Héra, K., Keleti, T., Máthé, A.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2106.09661
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⊂ 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 Borel set E⊂ ℝk× ℝn with dim (projℝk E)=k there is a Γk-null subset G⊂ E such that dim (E∖ G) = k+ess-sup(dim Et) where ess-sup(dim Et) is the essential supremum of the Hausdorff dimension of the vertical sections \Et\t∈ ℝk 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.