Lithuanian Mathematical Journal

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

On the exceptional sets in Sylvester expansions*

  • Meiying Lü


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 \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)


Sylvester expansion exceptional set Hausdorff dimension 


11K55 28A80 


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  1. 1.
    K. J. Falconer, Fractal Geometry, Mathematical Foundations and Application, JohnWiley & Sons, Chichester, 1990.zbMATHGoogle Scholar
  2. 2.
    J. Galambos, Representations of Real Numbers by Infinite Series, Lect. Notes Math., Vol. 502, Springer, Berlin, Heidelberg, 1976.CrossRefGoogle Scholar
  3. 3.
    Y.Y. Liu and J. Wu, Hausdorff dimensions in Engel expansions, Acta Arith., 99(1):79–83, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Y.Y. Liu and J. Wu, Some exceptional sets in Engel expansions, Nonlinearity, 16(2):559–566, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    B. Wang and J. Wu, The growth rates of digits in the Oppenheim series expansions, Acta Arith., 121(2):175–192, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. Wu, On the distribution of denominators in Sylvester expansions, Bull. Lond. Math. Soc., 34(1):16–20, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Wu, On the distribution of denominators in Sylvester expansions. II, Math. Proc. Camb. Philos. Soc., 135(3):421–430, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Wu, The Oppenheim series expansions and Hausdorff dimensions, Acta Arith., 107(4):345–355, 2003.MathSciNetCrossRefzbMATHGoogle Scholar

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© 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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