The growth rate of the partial quotients in a class of continued fractions with parameters

作者：Zhong, Ting^*; Tang, Liang

来源：Journal of Number Theory, 2014, 145: 388-401.

DOI：10.1016/j.jnt.2014.06.012

• GCF(is an element of) expansion
• Parameter function
• Engel series expansion
• Arithmetic properties
• Hausdorff dimension

Let is an element of : N -> R be a parameter function satisfying the condition is an element of(k) + k + 1 > 0 and let T-is an element of : (0,1] -> (0,1] be a transformation defined by
T-is an element of(x) = -1 + (k + 1)x/1+ is an element of(k) - k is an element of(k)x for x is an element of (1/(k + 1), 1/k].
Under the algorithm T-is an element of, every x is an element of (0,1] is attached an expansion, called generalized continued fraction expansion with parameters by F. Schweiger [3]. Define the sequence {k(n) (x)}(n) >= 1 of the partial quotients of x by k(1) (x) = left perpendicular 1/x right perpendicular and = k(n)(x) = k(1)(T-is an element of(n-1)(x)) for every n >= 2. It is clear that under the condition satisfied by the parameter function is an element of, k(n+1)(x) >= k(n) (x) for all n >= 1. In this paper, we consider the size of the set given by
E-is an element of(alpha) := {x is an element of (0, 1] : k(n+1) (x) >= k(n)(x)(alpha) for all n >= 1}
for any alpha >= 1. We show that
{1/alpha, when is an element of(k) equivalent to is an element of(0) (constant); dim(H) E-is an element of(alpha) = 1/alpha-beta+1, when is an element of(k) similar to k(beta) and alpha >= beta >= 1; 1, when is an element of(k) similar to k(beta) and alpha < beta.
where dim(H) denotes the Hausdorff dimension. The first result generalizes a result of J. Wu [5] who considered the case when is an element of equivalent to 0 (i.e., Engel expansion).

• 出版日期2014-12
• 单位吉首大学