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

Committee Chairperson

Dr. Jingcheng Tong

Second Advisor

Dr. Adel Boules

Rights Statement

http://rightsstatements.org/vocab/InC/1.0/

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.

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