Some Properties on the Error-Sum Function of Alternating Sylvester Series ()
1. Introduction
For any
, let
and
be defined as
(1)
where
denote the integer part. And we define the sequence
as follows:
(2)
where
denotes the nth iterate of
.
It is well known that from the algorithm (1), all
can be developped uniquely into an infinite or finite series
(3)
In the literature [2], (3) is called the Alternating Balkema-Oppenheim expansion of x and denoted by
for short. From the algorithm, one can see that T maps irrational element into irrational element, and the series is infinite. While for rational numbers, in fact, we have
is rational if and only if its sequence of digits
is terminate or periodic, see [1-3].
For any
and
, define
![](https://www.scirp.org/html/18-5300182\4bf44d23-e474-48eb-9ffd-cade1c2b7d41.jpg)
From the algorithm of (1), it is clear that
(4)
For any
, let
be its Alternating Sylvester expansion, then we have
for any
. On the other hand, any
of integer sequence satisfying
for all
is a Sylvester admissible sequence, that is, there exists a unique
such that
for all
, see [9].
The behaviors of the sequence
are of interest and the metric and ergodic properties of the sequence
and
have been investigated by a number of authors, see [1-3].
For any
, define
(5)
and we call
the error-sum function of Alternating Sylvester series. By (4), since
for all
, then
and
is well defined. In this paper, we shall discuss some basic nature of
, also the Hausdorff dimension of the graph of
is determined.
2. Some Basic Properties of ![](https://www.scirp.org/html/18-5300182\7cd215e3-a1b0-41cc-803e-e43c1382bdfa.jpg)
In what follows, we shall often make use of the symbolic space.
For any
, let
![](https://www.scirp.org/html/18-5300182\6aacbd88-0d55-48f6-85d1-a97e35e5c494.jpg)
Define
![](https://www.scirp.org/html/18-5300182\b63265d2-8722-4f87-a754-06edb1e42297.jpg)
For any
, write
(6)
(7)
We use
to denote the following subset of (0,1],
(8)
From theorem 4.14 of [8], we have
when
is even, and
when
is odd. Finally, define
(9)
Lemma 1. For any
and
1)
(10)
2)
(11)
3)
(12)
Proof. 1) Since
and
, so when
, we can get
![](https://www.scirp.org/html/18-5300182\ecb085d9-8229-453d-9f27-569122552fa8.jpg)
accordingly
![](https://www.scirp.org/html/18-5300182\beddc1df-e238-4251-bcb4-b212e1ddd346.jpg)
we write
, so
.
Now
implies
for ![](https://www.scirp.org/html/18-5300182\591a190a-e1d4-4d17-b63a-c63a9a5c933c.jpg)
Thus
![](https://www.scirp.org/html/18-5300182\16d5cc39-3eac-444b-9428-3cdc04b7780a.jpg)
let
, we have
and
, thus
![](https://www.scirp.org/html/18-5300182\a6f13735-e0a8-45dc-966c-56a4a1d06b3f.jpg)
2) From 1) we know that
![](https://www.scirp.org/html/18-5300182\94722da3-91ba-4991-a6f1-3f1a5c7710b3.jpg)
from the definition of
we also know that
, so ![](https://www.scirp.org/html/18-5300182\8f53c579-84f9-4483-9364-4cd89fde1086.jpg)
![](https://www.scirp.org/html/18-5300182\f21e3bd6-284d-4507-9866-f80a7b0f1e82.jpg)
thus
![](https://www.scirp.org/html/18-5300182\f345a308-26d8-4033-b9c5-f7091d58ad2a.jpg)
3) Since as
,
![](https://www.scirp.org/html/18-5300182\245d8111-b28b-4c54-85bb-15a0faf79a87.jpg)
Thus
![](https://www.scirp.org/html/18-5300182\fe3f621d-3113-4dcf-a88c-e56843c685ce.jpg)
Let
![](https://www.scirp.org/html/18-5300182\bcc3bc93-83a9-42b0-9830-1d1e988cefae.jpg)
Proposition 2. For any
, if
, then
is left continuous but not right continuous. If
, then
is right continuous but not left continuous.
Proof. For any
and
, write
,
, where
,
are given by (6) and (7).
Case I,
, then
(13)
(14)
and
. For any
, since when ![](https://www.scirp.org/html/18-5300182\d0a71dc0-5214-4599-af79-175c41fd8b35.jpg)
![](https://www.scirp.org/html/18-5300182\f1b305a2-4f1e-405b-8ac9-faa3d8ba704f.jpg)
This situation is included in Case II, so we can take
and
![](https://www.scirp.org/html/18-5300182\b43a0a14-a107-4ff0-be6c-720d0f7e6edf.jpg)
i.e.
![](https://www.scirp.org/html/18-5300182\60532db5-4b19-4306-981c-74363e17ef64.jpg)
![](https://www.scirp.org/html/18-5300182\b4a52f4e-246e-4ebc-abad-69f3e15729f7.jpg)
By (2),
![](https://www.scirp.org/html/18-5300182\a6f2d9d7-630f-468a-842c-315b8976621d.jpg)
which implies
![](https://www.scirp.org/html/18-5300182\b873637e-c63c-42e3-9ec8-4b67813e7cd2.jpg)
and
![](https://www.scirp.org/html/18-5300182\655f51dd-c2c0-4e7c-949b-f0ccbdac556d.jpg)
Let
, we get
and
, thus
![](https://www.scirp.org/html/18-5300182\27738de2-84ed-4f8c-9c6d-0d2976f65cfd.jpg)
and this implies
is left continuous at
.
Let
![](https://www.scirp.org/html/18-5300182\75e9d20f-dd73-4cc4-9825-278e9cd9b70c.jpg)
![](https://www.scirp.org/html/18-5300182\35821bcd-eecb-4210-a2ac-ad0fe5aad6ad.jpg)
then
![](https://www.scirp.org/html/18-5300182\02654ef8-82ee-4f07-9fff-c8f655ba9bcf.jpg)
Let
, we have
![](https://www.scirp.org/html/18-5300182\61f4626a-4ff8-48a9-95f8-67b0febb4cf8.jpg)
and this implies
is not right continuous at
. For
(15)
following the same line as above, we have
![](https://www.scirp.org/html/18-5300182\344635fd-ad9e-473a-9084-5f5aa7629461.jpg)
Case II ![](https://www.scirp.org/html/18-5300182\ff9b0ff3-0ff3-426b-a407-c6c11e95e737.jpg)
Let
(16)
(17)
Following the same line as above, we have
![](https://www.scirp.org/html/18-5300182\cc2adce5-cda7-4944-b2e8-ad0146c33abb.jpg)
![](https://www.scirp.org/html/18-5300182\b4240caa-ba03-4d69-af2e-358698fe0359.jpg)
and
is right continuous.
Corollary 3. For any
and
, write
,
. Then for any
, if
then
![](https://www.scirp.org/html/18-5300182\ce7f24e5-0cf2-4035-bf8f-5264433fc61e.jpg)
where
.
From the corollary, for any ![](https://www.scirp.org/html/18-5300182\d78af628-f505-41a2-a11a-67327ac2b353.jpg)
![](https://www.scirp.org/html/18-5300182\c6f7ebab-7609-4d52-bd4b-4cd96be64edf.jpg)
where
is the Lebesgue measure of
.
Theorem 4.
is continuous on
.
Proof: For any
and
, let
be its Alternating Sylvester expansion. For any
, write
. By (Corollary 3), for any
, we have
![](https://www.scirp.org/html/18-5300182\dfb92cba-09f9-486f-a16f-f6d5b0e58c46.jpg)
Write
, where
![](https://www.scirp.org/html/18-5300182\64b46c66-accf-4041-8cc6-9468aa7407eb.jpg)
Theorem 5. If
, then there exists
, such that ![](https://www.scirp.org/html/18-5300182\87447222-971d-4293-95b4-86ca0e91a869.jpg)
Proof. Set
, then
has the same continuity as
. Write
![](https://www.scirp.org/html/18-5300182\9e41cbb0-9b1c-4c7f-a271-b3f53c4d8b33.jpg)
trivially,
, then the set is well defined.
If
, then by the left continuity of
, we have
![](https://www.scirp.org/html/18-5300182\c9f5068c-9bab-4b24-b745-3e8bd2436fb2.jpg)
As a result, there exists a
such that for any
.
If
, since
is not left continuous, then
such that for any
,
, that is
.
Following the same line as above, we can prove
.
Now we shall prove that
. We can choose
such that
, if
, then
![](https://www.scirp.org/html/18-5300182\a4cc5ea0-1907-4a20-8895-b86f38471be1.jpg)
if
, then
![](https://www.scirp.org/html/18-5300182\2119a947-a89d-4be0-821b-964dd5dbec62.jpg)
In both case
. Following the same line as above, we can prove
, and
.
Therefore, there exists
, such that ![](https://www.scirp.org/html/18-5300182\38ea5bbd-fcc8-4aaf-ab88-2312ff75872a.jpg)
Theorem 6.
and ![](https://www.scirp.org/html/18-5300182\e4f6727a-8c52-4bee-bc90-0c907fe5f543.jpg)
Proof.
![](https://www.scirp.org/html/18-5300182\970bf36b-9a70-4745-a280-b167b5061760.jpg)
Let
, then
thus
![](https://www.scirp.org/html/18-5300182\b7c67c52-afd6-4c88-b452-27d8b4c42a97.jpg)
thus,
![](https://www.scirp.org/html/18-5300182\58b3bb11-5e5e-454e-b0f1-045cacacc800.jpg)
Through the MATLAB program we can get the definite integration
![](https://www.scirp.org/html/18-5300182\2c978b56-daa9-4460-9939-8fca7357a9f3.jpg)
3. Hausdorff Dimension of Graph for ![](https://www.scirp.org/html/18-5300182\79e2e257-7f44-45eb-b2eb-1978a3592b62.jpg)
Write
![](https://www.scirp.org/html/18-5300182\a21b2519-0daa-4452-9168-1fda6e69693e.jpg)
Theorem 7.
.
Proof. For any
,
is a covering of
. From (Cor 3),
can be covered by
squares with side of length
. For any
,
![](https://www.scirp.org/html/18-5300182\c3754810-37b0-485c-b030-749ddc159ab7.jpg)
Thus, ![](https://www.scirp.org/html/18-5300182\3ec35f0b-99f3-4baf-a012-56fa2f2e1d83.jpg)
Since
then
![](https://www.scirp.org/html/18-5300182\23806517-29c2-4421-8ca3-80072a718b5e.jpg)
so
.
4. Acknowledgements
This work is supported by the Hunan Education Department Fund (11C671).
NOTES