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The growth rates of digits in the Oppenheim series expansions

100%

Wang B.-W., Wu J.

Acta Arithmetica

2006

tom 121

nr 2

175-192

The growth speed of digits in infinite iterated function systems

81%

Cao C.-Y., Wang B.-W., Wu J.

Studia Mathematica

2013

tom 217

nr 2

139-158

Let ${fₙ}_{n≥1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${aₙ(x)}_{n≥1}$ of integers, called the digit sequence of x, such that $x = lim_{n→∞} f_{a₁(x)}∘ ⋯ ∘ f_{aₙ(x)}(1)$. We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set ${x ∈ Λ: aₙ(x) ∈ B (∀ n ≥ 1), lim_{n→∞} aₙ(x) = ∞}$ for any infinite subset B ⊂ ℕ, a question posed by Hirst for continued fractions. Also we generalize Łuczak's work on the dimension of the set {x ∈ Λ: $aₙ(x) ≥ a^{bⁿ}$ for infinitely many n ∈ ℕ} with a,b > 1. We will see that the dimension of the sets above is tightly connected with the convergence exponent of the contraction ratios of the sequence ${fₙ}_{n≥1}$.

Ograniczanie wyników

1 Acta Arithmetica

1 Studia Mathematica

2 Wang B.-W.

2 Wu J.

1 Cao C.-Y.

1 2013

1 2006

Wyniki wyszukiwania

The growth rates of digits in the Oppenheim series expansions

The growth speed of digits in infinite iterated function systems