Good’s theorem for Hurwitz continued fractions

Robert, Gerardo González

doi:10.1142/s1793042120500761

Cited by 4 publications

(3 citation statements)

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“…Analogues of Good's theorem [11] were obtained for other series expansions of numbers, see [10,26]. Taking B = N in Theorem B yields an extension of Good's theorem to the SRCF as follows.…”

Section: Andmentioning

confidence: 85%

“…Let ǫ ∈ (0, τ (B)). As in [31], we define a strictly increasing sequence {b m } m∈N in B inductively as follows: b 1 = min B, and b m+1 > b m is the minimal in B such that (10)…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…Set s ǫ,δ = (τ (B) − ǫ)/(2 + δ). By(10) we have 2+δ)s ǫ,δ ln ≥ Substituting s = s ǫ,δ and combining the above two inequalities yieldZ Φ n (s ǫ,δ ) ≥ K −N s ǫ,δ (min B) −2sǫ,δ t It follows that P Φ (s ǫ,δ ) = lim inf n→∞ (1/n) log Z Φ n (s ǫ,δ ) ≥ 0, and hence (17) s(Φ) ≥ τ (B) − ǫ 2 + δ .…”

mentioning

confidence: 99%

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Hausdorff dimension of sets with restricted, slowly growing partial quotients in the semi-regular continued fraction

Nakajima¹,

Takahasi²

2022

Preprint

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We consider sets of irrational numbers whose partial quotients a σ,n in the semi-regular continued fraction expansion obey certain restrictions and growth conditions. Our main result asserts that, for any sequence σ ∈ {−1, 1} N in the expansion, any infinite subset B of N and for any function f on N with values in [min B, ∞) and tending to infinity, the set of irrationals in (0, 1) such that a σ,n ∈ B, a σ,n ≤ f (n) for all n ∈ N and a σ,n → ∞ as n → ∞ is of Hausdorff dimension τ (B)/2, where τ (B) is the exponent of convergence of B. We also prove that for any σ ∈ {−1, 1} N and any B ⊂ N, the set of irrationals in (0, 1) such that a σ,n ∈ B for all n ∈ N and a σ,n → ∞ as n → ∞ is also of Hausdorff dimension τ (B)/2. To prove these results, we construct nonautonomous iterated function systems well-adapted to the given restrictions and growth conditions, and then apply the dimension theory developed by Rempe-Gillen and Urbański.

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“…Analogues of Good's theorem [11] were obtained for other series expansions of numbers, see [10,26]. Taking B = N in Theorem B yields an extension of Good's theorem to the SRCF as follows.…”

Section: Andmentioning

confidence: 85%

“…Let ǫ ∈ (0, τ (B)). As in [31], we define a strictly increasing sequence {b m } m∈N in B inductively as follows: b 1 = min B, and b m+1 > b m is the minimal in B such that (10)…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%