On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

Abstract

Abstract For a set A of positive integers a1 < a2 < · · ·, let d(A), d¯(A) $\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as λ(A)=lim⁡ supn→∞an+1an $\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$ . The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, d¯(A)=β $\overline d (A) = \beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping ϱ(A)=∑n=1∞χA(n)2n $\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $ , where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension dimϱ𝒢(α,β,γ)=min⁡{δ(α),δ(β),1γmax⁡σ∈[αγ,β]δ(σ)}, $$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$ where δ is the entropy function δ(x)=−x log2 x−(1−x) log2 (1−x). $$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$