Looking for analysis books with in-depth treatment of Cantor Set and Cantor Function

Question

I'm a third-year mathematics undergraduate student currently taking a second course in Real Analysis.

In the course, we were briefly introduced to the version of the Cantor Set being the set of all real numbers in $[0,1]$ that have ternary expansion containing no $1 \text{'s}$, and then from there we were given problems pertaining to the Cantor Set that we had to solve. Now, the issue I'm having is we were never formally introduced to what a $p \text{-ary}$ expansion of a real number actually is and how we actually work with them. Consequently, I am constantly struggling to even begin thinking of an approach to solve those problems. As for the Cantor Function, it is pretty much the same issue where we were simply given the basic definitions and then left alone to solve problems pertaining to it. The textbook that we are using is "An Introduction to Lebesgue Integration and Fourier Series" by Howard J. Wilcox and it isn't of much help either as it only has a mild treatment for the Cantor Set and has almost no treatment whatsoever for the Cantor Function.

So, with all that being said, I'm looking for resources (preferably textbooks or notes) that at least cover the following concepts about the Cantor Set and Cantor Function:

1. A proof that the elements of the Cantor Set can be given by such ternary expansion.
2. Some examples of how properties of the Cantor Function are proven using the ternary expansion formulation. For example:
   • How do we show that the Cantor Set is nowhere dense, perfect, etc.
   • How do we prove that a particular number is or is not in the Cantor Set.
   • How do we show that the Indicator Function of Cantor Set is/is not Riemann Integrable.
3. An overview of the Cantor Function and some of its alternate forms; e.g.
   • The standard form of the Cantor Function
   • The recursive form of the Cantor Function
4. Some examples of how the different forms of the Cantor Function are used prove facts about the Cantor Function. For example:
   • How do we determine the image of a particular number of the Cantor Function using the two formulation
   • How do we evaluate the continuity, uniform continuity, differentiability, and Riemann Integrability of the Cantor Function.

I've been looking around for quite some time now but I haven't had much luck. So, any suggestion would be greatly appreciated.

Answer 1

To get you started, here's an overview of the Cantor set's definition, and proofs of each property in your second point. Proofs relating to the Cantor function are very similar in nature, essentially reusing the same ideas.

1. I'm not entirely sure on what you meant in this part of your question, but I'll clarify precisely what "a ternary expansion containing no 1's" means. The ternary expansion of $x\in[0,1]$ is given by $x=\sum_{n=1}^\infty x_n3^{-n}$, where $a_n$ is a sequence uniquely determining $x$. We call $x_n$ the '$n$th digit' of $x$, and I'll refer to the sequence $x_n$ as the ternary expansion of $x$ in the rest of this answer. Note that $x$ has a unique ternary expansion unless there is an $N$ where $x_n=0$ for all $n>N$ or $x_n=2$ for all $n>N$, in much the same way that "$0.999\ldots=1$" in decimal expansions. (Exercise: verify this claim using geometric sums). To remove this ambiguity, one should define the Cantor set more carefully as the set of all $x\in[0,1]$ which have at least one ternary expansion without a 1. So by this definition, for example, $\frac13=\sum_{n=2}^\infty 2\cdot3^{-n}$ and $\frac23=\frac23+\sum_{n=2}^\infty0\cdot3^{-n}$ are both in the Cantor set, because they have an expansion containing no 1s, even though their other expansion does contain 1s. Similarly, 0 and 1 both end up in the Cantor set by this definition.

2. Closed: suppose $x$ is a limit point of $C$, but $x\notin C$, so that there is at least one $n$ where the $n$th digit $x_n$ is equal to 1. Take the smallest such $n$. There must also be some $k>n$ such that $x_k\neq0$, because otherwise, every digit past $x_n$ would be a zero, so $x$ would have an alternate ternary expansion without a 1, and hence would be in the Cantor Set. Similarly, there must be some $m>n$ such that $x_m\neq2$. Then, if $y\in(x_n-3^{-k-1},x_n+3^{-m-1})$, $y$ also has $y_n=1$ in its ternary expansion (why?), and so is not in the Cantor set. So $x$ has an open neighbourhood $B_x$ disjoint from $X$. Taking the union $\bigcup_{x\notin C}B_x$, we see that the complement $C^c$ of the Cantor set is open, so the Cantor set is closed.

• Perfect: Since we have shown $C$ is closed, it just remains to show that every point of $C$ is a limit point. Let $x\in C$, with a ternary expansion given by $(x_n)$. We construct a sequence $y(k)\in C$ by defining its ternary expansion as follows: if $n\leq k$ then $y(k)_n$ (the $n$th digit of $y(k)$) is equal to $x_n$; else, $y(k)_n=0$. Clearly $y(k)\in C$. Then $|y_k-x|<3^{k-1}$, so $y_k\to x$ and $x$ is a limit point of $C$.

• Nowhere dense: Since $C$ is closed, all we need to show is that it contains no interval. If $C$ contained an interval, it would contain a closed interval; so suppose for the sake of contradiction that $[x,y]\subset C$, $x<y$. Since $x\neq y$, they differ at some $n$th digit. Take the smallest such $n$. Then, since $x<y$ and these are both in $C$, we know $x_n=0$, $y_n=2$. If $x'_k=x_k$ for all $k\neq n$ and $x'_n=1$ is a ternary expansion for $x'$, then $x'\in[x,y]\setminus C$, a contradiction.

• $C$ can be shown to be uncountable by a diagonalisation argument. This is a little fiddly to carry out, and the well-known diagonalisation argument showing $\mathbb R$ is uncountable is identical up to labelling digits and being careful with uniqueness of expansions, so I won't replicate it here.

• There is another definition of $C$: Let $C_0=[0,1]$; let $C_1=[0,\frac13]\cup[\frac23,1]$; and so on, so that $C_{n}=\bigcup_{k=1}^{3^{n-1}-1}\left(\left[\frac{3k}{3^n},\frac{3k+1}{3^n}\right]\cup\left[\frac{3k+2}{3^n},\frac{3k+3}{3^n}\right]\right)$. Then $C=\bigcap_{n=0}^\infty C_k$. To see that this is an equivalent definition, note that if $x$ has no decimal expansion not containing a 1, and $n$ is the smallest $n$ with $x_n\neq1$, then $x\notin C_n$; and conversely if $x$ is not in $C_n$ for some $n$, which we pick to be the smallest such $n$ for convenience, then $x_n=1$ (why?).

• The indicator function $\mathbf1_C$ is in fact Riemann-integrable; the lower integral $\underline\int_0^1\mathbf1_C=0$, by nowhere-density; and the upper integral can be shown to be zero by considering the integral of a certain sequence of step functions, constructed using the recursive characterisation of $C$. In particular, this means $C$ has measure zero.

Answer 2

[1] Cantor Set: [1-1] Soltanifar, Mohsen (2006) "On A Sequence of Cantor Fractals," Rose-Hulman Undergraduate Mathematics Journal: Vol. 7 : Iss. 1 , Article 9. Available at: https://scholar.rose-hulman.edu/rhumj/vol7/iss1/9 [1-2] Soltanifar, M. A Different Description of A Family of Middle-α Cantor Sets. Am. J. Undergrad. Res. 2006, 5, 9–12. Available at: https://www.ajuronline.org/uploads/Volume%205/Issue%202/52C-SoltanifarArt.pdf [1-3] Soltanifar, M. A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals. Mathematics 2021, 9, 1546. Available at: https://doi.org/10.3390/math9131546 [1-4] Soltanifar, M. The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals. Mathematics 2022, 10, 706. https://doi.org/10.3390/math10050706

[2] Cantor Function: [2-1] Soltanifar, M. A Classification of Elements of Function Space F(R,R). Mathematics 2023, 11, 3715. Available at: https://doi.org/10.3390/math11173715