Elementary proof of the estimation of the sum of fractional parts
Author(s):
A.V. Shutov, candidate of Sciences, associate Professor, Vladimir State University named after Alexander and Nikolay Stoletovs, Vladimir, Russia,
a1981@mail.ru Issue:
Volume 50, № 4
Rubric:
Physics. Mathematical modeling
Annotation:
Let be an irrational number. Denote by { } n q a sequence of partial quotients of the continued fraction
expansion of and by { } n
n
P
Q
a sequence of partial convergents to . Assume that
1
0
1
( , ) ({ } )
2
n
n
i
С i
. Estimates for ( , ) n С are important both by themselves and in
connection with their applications in a some number-theoretic problems, such that studying the remainder
term of the distribution of the fractional parts of a linear function and in number-theoretic methods of
approximate integration. The best known estimate of ( , ) n С is
1
| ( , ) |
k
n i
i
С C q
for
1 k i Q . In this paper, we give a new short proof of this result.
Keywords:
sums of fractional parts, uniform distribution, continued fractions.
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