Dimension of the Non-Differentiability Subset of the Cantor Function ()
1. Introduction
Let T be the Cantor middle thirds set (Start with the closed interval
. Remove the open interval
to obtain
, i.e. two disjoint
closed segments. Remove the middle thirds of those two segments, and you end up with four disjoint segments. After infinitely many steps, the result is called the Cantor set.) and
be the Cantor ternary function (The function can be defined in the following way, given
consider its ternary expansion
, i.e. a choice of coefficients
such that
(1.1)
Define
to be
if none of the coefficients
takes the value 1 and the smallest integer n such that
otherwise.
Then
(1.2)
Let
be the set of non-differentiable points of Cantor function
.
Denote
(1.3)
and
(1.4)
See [1], we have
(1.5)
So,
Let
be the ternary expansion of a point in T, which is not an endpoint of a complementary interval. Let
denote the position of the nth zero in this ternary expansion, and
denote the position of the nth two in the expansion of t.
For each
and
, define
Firstly, several authors [2] [3] [4] have proved that the Hausdorff dimension of the set
is
and the packing dimension of
is
.
Let the Cantor T in R be defined by
with a disjoint union, where
the
are similitude mappings with ratios
. Let
be the self-similar Borel probability measure on T corresponding to the probability vector
. Let S be the set of points at which the probability distribution function
of
has no derivative, finite or infinite. On the one hand, for the case where
, Wen xia [5] has been studied the packing and box dimensions of S and give an approach to calculate the Hausdorff dimension of S. Kennen J. Falconer [6] is further noted that Hausdorff dimension of the set of points of non-differentiability of a self-affine “devil’s staircase” function is the square of the dimension of the set of points of increase. On the other hand when
, for the case where
or
. Yuanyuan yao, YunXin zhang and wenxia [1] Systematic has been given the Hausdorff and packing dimension of S. Secondly, Reza Mirzaire [7] find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, he show that the boundary of a convex open subset of
,
, has Hausdorff dimension at most
and David Pavlica [8] characterize sets of non-differentiability points of convex functions on
. This completes (in
) the result by [9] which gives a characterization of the magnitude of these sets. Last, Eidswick [10] points out that the points of nondifferentiability of the Cantor ternary function are characterized in terms of the spacing of 0’s and 2’s in ternary expansions, and have calculated that
has the cardinality of the continuum.
Since about the dimension of the set
of points, Eidswick didn’t study it. So, our main job in this paper is to give its dimension, that is, the following theorem.
Theorem 1.1.
, and
.
2. Proofs of Theorem
First, we give a proposition, which simplifies the calculation of
and
.
Proposition 2.1. Let
be ternary expansion of t, and define
(2.1)
(2.2)
Let
we have
(2.3)
and
(2.4)
Proof. We just need to prove (2.3), since (2.4) is proved in a similar way.
If
then, the proposition is apparently true.
So, we suppose that
then, we can find
such that
(2.5)
Note that
and
, we have
Therefore, for any
, there is an integer
, such that
.
By (2.5), for any
, there is an integer
such that: when
,
Denote
, then
, we have
So,
and since
, it follows that
On the other hand, it is obvious that:
so we complete the proof of (2.3).
☐
Besides, we need the following Lemma due to Billingsley:
Lemma 2.2. Let
be an integer and for
, let
be the nth generation half-open b-adic interval of the form
cintaining x, let
be Borel and let
be a finite Borel measure on [0, 1], suppose
. If
then
For the proof of this lemma, we refer readers to Section 1.4 of [11].
Corollary 2.3. For
, define
(2.6)
We have
(2.7)
Proof. To see this, let
be the probability measure on
that give equal measure to nth generation covering intervals. This measure makes the digits
in (2.6) independent identically distributed uniform random bits. For any
,
Thus the liminf of the left-hand side is the liminf of the right-hand side. By Lemma 2.2, this proves the corollary.
Last, we need to introduce a theorem and a formula:
Theorem 2.4. Let t be a point of T which is not an endpoint of a comple-mentary interval, let
denote the position of the nth zero in its ternary
expansion, and let
. Then
. Furthermore, if
, then,
.
Let,
. Then,
(2.8)
The proof of theorem 2.4 and formula (2.8), we see [10].
Proof of Theorem. First, let’s construct a subset of
.
Let
,
. We can find increasing sequences of positive integers
and
such that
(2.9)
and
(2.10)
when n is large enough, where
. At the same time, we may request that
,
and
hold for all
. Moreover, denote
and
, we insert these points
and
in to
and
, respectively, such that
(2.11)
and
(2.12)
when n is large enough, for
and
; and let
,
. We will show the selection of
and
later.
Define
, if
, or
.
Define
, if
, or
.
And restrict
(That is,
is free to take a value of 0 or 2 for all other values of m).
And we claim that
. We’ll give a simple proof after the selection of
and
, let
be the set of all such t.
For the selection of
,
, first of all, take
. Assume that
and
is taken, we choose
, such that
(a)
;
(b)
(note that
);
(c)
(note that
is uniformly distributed modulo 1).
where
represents the integer part that represents taking x, and
represents the fractional portion of the y. And take
. We choose
such that
(d)
;
(e)
;
(f)
.
and take
.
These
and
meet the desired conditions.
From
and
, we can get that
To see that
, by the Theorem 2.4 and formula (2.8), we only
need to show that
and
are adjacent positions of zeros, whose spacing is largest; and
and
are adjacent positions of twos, whose spacing is largest.
Since
, for any
, when n is large enough, we have
then, when n is large enough
Then, according to our above-mentioned construction of
, other
spacings of zeros are at most 2n, so
is the largest spacing of zeros. Similarly, we have
, and
is the largest spacing of twos.
Denote
Let
in fact, the set B is all positions that
can freely select 0 or 2 in the above construction. Using Corollary 2.3, we have
and
And we calculate these as follows: It’s not hard to see that, when
, we can get the superior limit:
where
,
.
On the other hand, when
, we can get the inferior limit:
and by calculating
where
,
and
;
where
,
,
.
In Section 1, we already know
, and
.
Since
, we have
and
.
3. Conclusion
Finally, through the above proof, we solved dimension of the set
of points that is Eidswick didn’t study it. In other words, we finally get that
, and
.