We prove that for any 1 ≤ k < n and s≤ 1 , the union of any nonempty s -Hausdorff dimensional family of k -dimensional affine subspaces of R^n has Hausdorff dimension k + s . More generally, we show that for any 0 < α ≤ k , if B ⊂ R^n and E is a nonempty collection of k -dimensional affine subspaces of R^n such that every P ∈ E intersects B in a set of Hausdorff dimension at least α , then dim B ≥ 2 α - k + min(dim E, 1) , where dim denotes the Hausdorff dimension. As a consequence, we generalize the well-known Furstenberg-type estimate that every α -Furstenberg set has Hausdorff dimension at least 2 α ; we strengthen a theorem of Falconer and Mattila [5]; and we show that for any 0 ≤ k < n , if a set A ⊂ R^n contains the k -skeleton of a rotated unit cube around every point of R^n , or if A contains a k -dimensional affine subspace at a fixed positive distance from every point of R^n , then the Hausdorff dimension of A is at least k + 1 .
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Héra et al. (2019) studied this question.