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
Define to be if none of the coefficients takes the value 1 and the smallest integer n such that otherwise.
Let be the set of non-differentiable points of Cantor function .
See , we have
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    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  has been studied the packing and box dimensions of S and give an approach to calculate the Hausdorff dimension of S. Kennen J. Falconer  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  Systematic has been given the Hausdorff and packing dimension of S. Secondly, Reza Mirzaire  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  characterize sets of non-differentiability points of convex functions on . This completes (in ) the result by  which gives a characterization of the magnitude of these sets. Last, Eidswick  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
Proof. We just need to prove (2.3), since (2.4) is proved in a similar way.
then, the proposition is apparently true.
So, we suppose that
then, we can find such that
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
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
For the proof of this lemma, we refer readers to Section 1.4 of .
Corollary 2.3. For , define
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,
The proof of theorem 2.4 and formula (2.8), we see .
Proof of Theorem. First, let’s construct a subset of .
Let , . We can find increasing sequences of positive integers and such that
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
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
(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
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.
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 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 .
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 .
 Bandt, C. and Graf, S. (1992) Self-Similar Sets. VII. A Characterization of Self-Similar Fractals with Positive Hausdorff Measure. Proceedings of the American Mathematical Society, 114, 995-1001.
 Darst, R. (1993) The Hausdorff Dimension of the Non-Differentiability Set of the Cantor Function Is [ln2/ln3]2. Proceedings of the American Mathematical Society, 119, 105 108.
 Eidswick, J.A. (1974) A Characterization of the Nondifferentiability Set of the Cantor Function. Proceedings of the American Mathematical Society, 42, 214-217.