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

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

“…Using Theorem 1.5 (or more precisely, a technical variant of the theorem), we obtain explicit estimates on the size of c(α). This estimate is stronger than what would be obtained by merely computing the value of c(α) from the arguments in [12].…”

“…While the authors in [12] do not explicitly compute c(α), the value obtained by their arguments would be extremely small. Using Theorem 1.5 (or more precisely, a technical variant of the theorem), we obtain explicit estimates on the size of c(α).…”

“…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).…”

“…With these definitions, we recall the following definition from [12]. (ii) For each l ∈ L, P l ⊂ l is a (δ, α) 1 -set (with implicit parameters ε and C), and #P l ≥ C −1 δ ε−α .…”

“…The authors in [12] proved Theorem 1.7 by showing that every discretized (α, 2α)-Furstenberg set must have large δ covering number. The relationship between these two statements is as follows.…”