An improved bound for the dimension of $(\alpha,2\alpha)$-Furstenberg sets

Abstract

We show that given ˛2 .0; 1/ there is a constant c D c.˛/ > 0 such that any planar .˛; 2˛/-Furstenberg set has Hausdorff dimension at least 2˛C c. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps.

An Incidence Estimate and a Furstenberg Type Estimate for Tubes in $$\mathbb {R}^2$$

“…The (δ, α) 1 spacing condition roughly says that #Y (T ) ∼ δ −α and for any δ × w-subtube T w ⊂ T ∩[0, 1] 2 there holds #{B δ ∈ Y (T ) : B δ ∩ T w = ∅} (w/δ) α . Our tube set T also satisfies an evenly spacing condition which is stronger than the (δ, β) 2 spacing condition: |T| ∼ δ −β (one may think δ −β = X W ); any w × 1 tube in [0, 1] 2 contains (w/δ) β many tubes in T. It was shown in [6] Lemma 3.3 that the δ-discretized version under the (δ, β) 2 -condition for T and (δ, α) 1 -condition for Y (T ) will imply the original Furstenberg problem (in terms of Hausdorff dimension).…”

“…Recently, Orponen and Shmerkin [10] further improved the bound to dim H E ≥ 2α + for α ∈ ( 1 2 , 1). A general Furstenberg set problem was also considered by many authors, for example in [6,8,10].…”

Additive properties of fractal sets on the parabola

“…This is a recent result of the first author with Shmerkin [17]. In the present paper, we only need the special case t " 2s, which was known much earlier: the case s " 1 2 is due to Bourgain [2] (modulo a slightly different definition of discretised Furstenberg sets), and the general case s P p0, 1q is due to Héra, Shmerkin, and Yavicoli [10]. In the case t " 2s, the best known constant " " is actually fairly large, due to recent work of Di Benedetto and Zahl [6].…”

“…In fact, we will prove Theorem 1.1 by applying the non-existence of ps, 2sq-Furstenberg sets of dimension 2s. The connection between the Furstenberg set problem and the Borel subring problem was initially discovered by Katz and Tao [11], and has been thereafter applied several times to make progress in the Furstenberg set problem, see [6,10,15,17]. In the current paper, we may use the best current estimates on ps, 2sq-Furstenberg sets as a black box.…”

On exceptional sets of radial projections

“…The main point is the following: it is known that the Hausdorff dimension of every ps, tq-Furstenberg set F Ă R 2 satisfies dim H F ě γps, tq, where γps, tq is the function defined in (3.7). The case t ď s is due to Lutz and Stull [15]; they used information theoretic methods, but a more classical proof is also available, see [8,Theorem A.1]. The case t ě s essentially goes back to Wolff in [25], but also literally follows from [8, Theorem A.1].…”