Lithuanian Mathematical Journal

, Volume 58, Issue 1, pp 48–53 | Cite as

On the exceptional sets in Sylvester expansions*

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Abstract

For any x 𝜖 (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {d j (x), j ≥ 1} is a sequence of positive integers satisfying d1(x) ≥ 2 and dj + 1(x) ≥ d j (x)(d j (x) − 1) + 1 for j ≥ 1. Suppose ϕ : ℕ → ℝ+ is a function satisfying ϕ(n+1) – ϕ (n) → ∞ as n → ∞. In this paper, we consider the set
$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$

and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)

Keywords

Sylvester expansion exceptional set Hausdorff dimension 

MSC

11K55 28A80 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesChongqing Normal UniversityChongqingChina

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