Improved Versions of Some Furstenberg Type Slicing Theorems for Self-Affine Carpets

Abstract

Let $F$ be a Bedford–McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$\begin{align*} & \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace\end{align*}$$and $$\begin{align*} & \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace,\end{align*}$$where $\dim ^*$ is Furstenberg’s star dimension (maximal dimension of microsets). This improves the state-of-the-art results on Furstenberg type slicing Theorems for affine invariant carpets.

An upper bound for the intermediate dimensions of Bedford–McMullen carpets

“…The general lower bound of [2,14] is a concave function between .0; 0/ and .1; dim B ƒ/, which enables to construct an example which shows that dim  ƒ is not convex in general; see Fig- In Figure 4.1, the (orange) plot depicting the upper bound comes from Theorem 1.2, while the (blue) plot depicting the lower bound comes from a combination of results in [2,14,22]. The ratio log n m has an important role in projection and slicing results about Bedford-McMullen carpets [1,15] and the Assouad spectrum has a phase transition here [18]. These together with Claim 2.2 strongly suggest to us that dim  ƒ may have a phase transition at log n m. Can dim  ƒ have additional phase transitions at other integer powers of log n m?…”

How much can heavy lines cover?

“…Let 𝑐 > 0 be an absolute constant to be determined a little later. We apply Proposition 2.1 at scale Δ with parameters 𝜖 = 𝜂 2 and 𝐴 ∶= 𝑐𝑆 and to the families {𝐵 𝑝 ∶ 𝑝 ∈  Δ } and {𝕋 𝑇 ∶ 𝑇 ∈  Δ } (recall that ( Δ ,  Δ ) is defined using these balls and ordinary tubes). The proposition requires 𝐴 ∈ [Δ −𝜂 2 , Δ −1 ], and this is satisfied because 𝐴 ⩽ 𝑆 = 𝛿 −𝜂 ⩽ Δ −1 .…”

“…Theorem 5.2. For every 𝑡 ∈ (1,2] and 𝑠 ∈ [𝑡 − 1, 1 3 (2𝑡 − 1)] there exists a compact set  ⊂ (2, 1) consisting of non-vertical lines, and with the following properties.…”

“…Using this information, we can make progress on the problem of heavy lines: Proposition 1.1. Let 𝐴 ⊂ ℝ 2 be a set with dim H 𝐴 = 𝑡 ∈ [1,2]. If 𝑠 > 1 3 (2𝑡 − 1), then (𝐴, 𝑠, 𝑒) = ∅ for almost all 𝑒 ∈ 𝑆 1 , therefore 𝔥(𝐴, 𝑠) = 0.…”

“…Let 𝐴 ⊂ ℝ 2 be a Borel set with dim H 𝐴 = 𝑡 ∈ [1,2]. If 𝐴 is Ahlfors-regular, (1.2) states that 𝔥(𝐴) ⩽ 1.…”