Advances in Pure Mathematics
Vol.2 No.6(2012), Article ID:25018,5 pages DOI:10.4236/apm.2012.26070
Some Properties on the Error-Sum Function of Alternating Sylvester Series
Science College of Hunan Agricultural University, Changsha, China
Email: Huiping_J@126.com, *lum_s@126.com
Received July 11, 2012; revised September 21, 2012; accepted September 29, 2012
Keywords: Alternating Sylvester Series; Error-Sum Function; Hausdorff Dimension
ABSTRACT
The error-sum function of alternating Sylvester series is introduced. Some elementary properties of this function are studied. Also, the hausdorff dimension of the graph of such function is determined.
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
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
In what follows, we shall often make use of the symbolic space.
For any, let
Define
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 and1) (10)
2) (11)
3) (12)
Proof. 1) Since and , so when, we can get
accordingly
we write, so.
Now implies
for
Thus
let, we have and, thus
2) From 1) we know that
from the definition of we also know that, so
thus
3) Since as,
Thus
Let
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
This situation is included in Case II, so we can take and
i.e.
By (2),
which implies
and
Let, we get and, thus
and this implies is left continuous at.
Let
then
Let, we have
and this implies is not right continuous at. For
(15)
following the same line as above, we have
Case II
Let
(16)
(17)
Following the same line as above, we have
and is right continuous.
Corollary 3. For any and, write,. Then for any, if then
where.
From the corollary, for any
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
Write, where
Theorem 5. If, then there exists, such that
Proof. Set, then has the same continuity as. Write
trivially, , then the set is well defined.
If, then by the left continuity of, we have
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
if, then
In both case. Following the same line as above, we can prove, and .
Therefore, there exists, such that
Theorem 6. and
Proof.
Let, then thus
thus,
Through the MATLAB program we can get the definite integration
3. Hausdorff Dimension of Graph for
Write
Theorem 7..
Proof. For any, is a covering of. From (Cor 3), can be covered by squares with side of length. For any,
Thus,
Since
then
so.
Figure 1. The graph of S(x).
4. Acknowledgements
This work is supported by the Hunan Education Department Fund (11C671).
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NOTES
*Corresponding author.