Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces

Abstract

We show that if B ⊂ R n and E ⊂ A(n, k) 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 with k − 1 < α ≤ k, then dim B ≥ α + dim E/(k + 1), where dim denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every α-Furstenberg set in the plane has Hausdorff dimension at least α + 1/2.More generally, we prove that if B and E are as above with 0 < α ≤ k, then dim B ≥ α + (dim E − (k − α )(n − k))/( α + 1). We also show that this bound is sharp for some parameters.As a consequence, we prove that for any 1 ≤ k < n, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of R n has Hausdorff dimension at least k + s k+1 .

Large sets with small injective projections

“…Since the objective is minimizing the size of B, it is not clear whether these intersections or collections of parallel k-planes are an important component of the construction. As an application of Corollary 1.3, we present constructions corresponding to those in [8] and [5], with the additional property that they consist of disjoint, nonparallel k-planes. We found in Theorem 1.1 that requiring injectivity of a continuous function will not necessarily reduce the Hausdorff dimension of its graph; here we find an analogous statement, that requiring k-planes to be disjoint and non-parallel does not necessarily increase the size of their union.…”

“…In this section we prove Theorem 1.4, which consists of modifications of constructions given in [5] and [8]. In both cases we present constructions with the same Hausdorff dimension as those previously presented, with the additional property that the k-planes used are disjoint and non-parallel (whereas in [5] and [8] they were not).…”

“…Proof of Theorem 1.4, part (ii). We modify the construction given in [5] to select only k-planes which are disjoint and nonparallel. Set m = s/(k + 1) .…”

New estimates on the size of $(α,2α)$-Furstenberg sets

“…For this example, (11) (and hence (10)) is true with C 1 = 3/α and C 2 = 3. Indeed, by (3), #I(l) ≥ δ ε−α V. Once p ∈ I(l) has been chosen, there are ≥ δ ε−α V elements q ∈ I(l).…”

“…Numerous authors [5,10,11,12,16,17,18] have obtained lower bounds for γ(α, β) in different regimes. The most relevant for our discussion is the following result of Héra, Shmerkin, and Yavicoli, which used a variant of Theorem 1.4 due to Bourgain [2] to obtain bounds on γ(α, 2α).…”