Year
1994
Paper Type
Master's Thesis
College
College of Arts and Sciences
Degree Name
Master of Science in Mathematical Sciences (MS)
Department
Mathematics & Statistics
First Advisor
Dr. Jingcheng Tong
Second Advisor
Dr. Adel Boules
Third Advisor
Dr. Peter Braza
Abstract
Let A = {an}∞n = 1 be a sequence of positive integers. There are two related sequences Pn and Qn obtained from A by taking partial convergents out of the number [0; a1, a2, ..., an, ...], where Pn and Qn are the numerators and denominators of the finite continued fraction [0; a1, a2, ...,an].
Let P(n) be the largest positive integer k , such that Pk ≤ n. The sequence Q(n) is defined similarly.
- • A known result of Barnes' Theorem states that
P
- (
n
- ) =
o
- (
n
- ) and
Q
- (
n
- ) =
o
- (
n
- ).
- • In this paper we improve this result as
P
- (
n
- ) =
O
- (log n) and
Q
- (
n
- ) =
O
- (log n), where it follows that
P
- (
n
- )=
o
- (
nε
- ) and
Q
- (
n
- ) =
o
- (
nε
- ) for any
ε
- >0.
Suggested Citation
Vafabakhsh, Seyed J., "Density of the Numerators or Denominators of a Continued Fraction" (1994). UNF Graduate Theses and Dissertations. 80.
https://digitalcommons.unf.edu/etd/80