Hausdorff dimension and projections related to intersections

Abstract

For Sg(x, y) = x − g(y), x, y ∈ R n , g ∈ O(n), we investigate the Lebesgue measure and Hausdorff dimension of Sg(A) given the dimension of A, both for general Borel subsets of R 2n and for product sets.

A Survey on the Hausdorff Dimension of Intersections

“…This was proven in [26]. The paper also contains dimension estimates for S g (A) when dim A ≤ n + 1 and estimates on the dimension of exceptional sets of transformations g. In particular…”

“…The case dim A > (n + 1)/2 is a special case of Theorem 7; recall (29). The proof of the case dim A + (n − 1) dim B/n > n is based on the spherical averages and the first estimate of (26). Here is a sketch.…”

“…We now discuss further the projections S g ; recall (19). They are particular cases of restricted projections, which recently have been studied extensively; see [2] (Section 5.4), and [25,26] and the references given there. Restricted means that we are considering a lower-dimensional subspace of the Grassmannian G(2n, n).…”

A survey on the Hausdorff dimension of intersections

“…(8.1) D(S g# (µ × ν))(z) 2 dL n z dθ n g = c δ(µ)(t)δ(ν)(t)t 1−n dt, at least if µ and ν are smooth functions with compact support, see [M9,Section 5.2].…”

Hausdorff Dimension Exceptional Set Estimates for Projections, Sections and Intersections

“…All we need to do is to check the estimate (3.3), after that the proof runs as that of [M5,Theorem 4.1]. The qualitative version of (3.3) was given in the proof of [M6,Theorem 4.2]. We just have to check that that argument yields the upper bound we need.…”