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Approximation by Egyptian Fractions and the Weak Greedy Algorithm
Authors
Hung Viet Chu
Publication date
30 May 2023
Publisher
View
on
arXiv
Abstract
Let
0
<
θ
⩽
1
0 < \theta \leqslant 1
0
<
θ
⩽
1
. A sequence of positive integers
(
b
n
)
n
=
1
∞
(b_n)_{n=1}^\infty
(
b
n
​
)
n
=
1
∞
​
is called a weak greedy approximation of
θ
\theta
θ
if
∑
n
=
1
∞
1
/
b
n
=
θ
\sum_{n=1}^{\infty}1/b_n = \theta
∑
n
=
1
∞
​
1/
b
n
​
=
θ
. We introduce the weak greedy approximation algorithm (WGAA), which, for each
θ
\theta
θ
, produces two sequences of positive integers
(
a
n
)
(a_n)
(
a
n
​
)
and
(
b
n
)
(b_n)
(
b
n
​
)
such that a)
∑
n
=
1
∞
1
/
b
n
=
θ
\sum_{n=1}^\infty 1/b_n = \theta
∑
n
=
1
∞
​
1/
b
n
​
=
θ
; b)
1
/
a
n
+
1
<
θ
−
∑
i
=
1
n
1
/
b
i
<
1
/
(
a
n
+
1
−
1
)
1/a_{n+1} < \theta - \sum_{i=1}^{n}1/b_i < 1/(a_{n+1}-1)
1/
a
n
+
1
​
<
θ
−
∑
i
=
1
n
​
1/
b
i
​
<
1/
(
a
n
+
1
​
−
1
)
for all
n
⩾
1
n\geqslant 1
n
⩾
1
; c) there exists
t
⩾
1
t\geqslant 1
t
⩾
1
such that
b
n
/
a
n
⩽
t
b_n/a_n \leqslant t
b
n
​
/
a
n
​
⩽
t
infinitely often. We then investigate when a given weak greedy approximation
(
b
n
)
(b_n)
(
b
n
​
)
can be produced by the WGAA. Furthermore, we show that for any non-decreasing
(
a
n
)
(a_n)
(
a
n
​
)
with
a
1
⩾
2
a_1\geqslant 2
a
1
​
⩾
2
and
a
n
→
∞
a_n\rightarrow\infty
a
n
​
→
∞
, there exist
θ
\theta
θ
and
(
b
n
)
(b_n)
(
b
n
​
)
such that a) and b) are satisfied; whether c) is also satisfied depends on the sequence
(
a
n
)
(a_n)
(
a
n
​
)
. Finally, we address the uniqueness of
θ
\theta
θ
and
(
b
n
)
(b_n)
(
b
n
​
)
and apply our framework to specific sequences.Comment: 14 pages, to appear in Indag. Math. (N.S.
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oai:arXiv.org:2302.01747
Last time updated on 02/03/2023