Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results

Abstract

In this survey we collect and discuss some recent results on the so called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dimH) of the sets in the Fα-class: for a given α ∈ (0, 1], a set E ⊆ R2 is in the Fα-class if for each e ∈ S there exists a unit line segment `e in the direction of e such that dimH(` ∩ E) ≥ α. For α = 1, this problem is essentially equivalent to the “Kakeya needle problem”. Define γ(α) = inf {dimH(E) : E ∈ Fα}. The best known results on γ(α) are the following inequalities: max {1/2 + α; 2α} ≤ γ(α) ≤ (1 + 3α)/2. In this work we approach this problem from a more general point of view, in terms of a generalized Hausdorff measure Hh associated with the dimension function h. We define the class Fh of Furstenberg sets associated to a given dimension function h. The natural requirement for a set E to belong to Fh, is that Hh(`e ∩ E) > 0 for each direction. We generalize the known results in terms of “logarithmic gaps” and obtain analogues to the estimates given above. Moreover, these analogues allow us to extend our results to the endpoint α = 0. For the upper bounds we exhibit an explicit construction of Fh-sets which are small enough. To that end we adapt and prove some results on Diophantine Approximation about the the dimension of a set of “well approximable numbers”. We also obtain results about the dimension of Furstenberg sets in the class Fαβ, defined analogously to the class Fα but only for a fractal set L ⊂ S of directions such that dimH(L) ≥ β. We prove analogous inequalities reflecting the interplay between α and β. This problem is also studied in the general scenario of Hausdorff measures.