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| 摘要: |
| 对于任意实数 x∈(0,1], 记 x=[d1,d2,…]为x的Sylvester连分数展式, 令ψ(n)为N上的正函数,本文研究了集合A(ψ):A(ψ)={x∈(0,1]:limn→∞ log dn(x)/ψ(n)=1}的Hausdorff维数. 通过构造覆盖和合适的Cantor型子集,我们得到了该集合的精确维数为dimH A(ψ)=lim infn→∞ ψ(1)+ψ(2)+…+ψ(n)/ψ(n+1).同时, 本文还考虑了Sylvester连分数展式的部分商满足log dn(x)/ψ(n)的极限是零或无穷时的集合的Hausdorff维数. |
| 关键词: Sylvester连分数 Hausdorff维数 增长速度 |
| DOI： |
| 分类号:O156.7 |
| 基金项目:重庆市教育委员会科学技术研究项目资助(KJQN202100528);重庆市自然科学基金资助(CSTB2022NSCQ-MSX1255). |
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| HAUSDORFF DIMENSIONS OF CERTAIN SETS IN TERMS OF THE SYLVESTER CONTINUED FRACTION EXPANSIONS |
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LIAO Xu
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| Abstract: |
| For x∈(0,1], let x=[d1,d2,…] be its Sylvester continued fraction expansions, we calculate the Hausdorff dimension of the set A(ψ) defined in terms of the Sylvester continued fraction expansions as A(ψ)={x∈(0,1]:limn→∞ log dn(x)/ψ(n)=1} where ψ(n) is a positive function defined on N. By constructing the covering and a suitable subset of Cantor, we get the exact Hausdorff dimension of the set as dimH A(ψ)=lim infn→∞ ψ(1)+ψ(2)+…+ψ(n)/ψ(n+1). At the same time, we also calculate the Hausdorff dimension of the set of points with limn→∞ log dn(x)/ψ(n)=0 or ∞. |
| Key words: Sylvester continued fraction expansions Hausdorfi dimension growth rate |
