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
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 
and
1)
(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
.
4. Acknowledgements
This work is supported by the Hunan Education Department Fund (11C671).
NOTES