A survey on the Hausdorff dimension of intersections

Abstract

Let A and B be Borel subsets of the Euclidean n-space with dim A + dim B > n. This is a survey on the question: what can we say about the Hausdorff dimension of the intersections A∩(g(B)+z) for generic orthogonal transformations g and translations by z.

Hausdorff dimension of plane sections and general intersections

“…Theorem 1.3 follows immediately from Theorem 3.1 once the product measure version (3.4) of (1.4) is verified for the related projections 𝑃 g , 𝑃 g (𝑥, 𝑦) = 𝑥 − g(𝑦), 𝑥, 𝑦 ∈ ℝ 𝑛 , g ∈ 𝑂(𝑛), see Section 4. This was already observed in the arXiv version of [13], but not in the journal.…”

“…See [10, chapter 7] and [13] for discussions and references for the intersection problems. I would like to thank the referee for useful comments.…”