(1a) Show that

• [5/7]₂ = .101(101) (Notation: (abc) means abcabcabc....)
• [5/8]₃ = .12(12)

(1b) What is [1/6]₂? (Hint; it begins with .00 and then there is a repeating pattern of length 2.) What is [1/6]₃?

(1c) Find [23/27]₃ and [7/5]₃ to 4 places (i.e., the first 4 digits in their ternary expansions).

You may want to look at the formula page for a description of how to compute binary and ternary expansions.

(2a) Indicate on a sketch of the interval [0,1] the subinterval of all numbers whose ternary expansion begins with .201 (see Figure 2.10 in the text).

(2b) Indicate on your sketch the point that has ternary expansion .111(111). What is this number?

Use the identification between the Cantor set and [0,1] discussed in class (and in the text on page 75) to find the numbers in the Cantor set that are matched with the following numbers from [0,1]; 0, 1, 3/4, 1/8, 3/8, 1/6. (The graph of this function is displayed on the weekly summary page.)

Explain why the function g which maps the interval [0,1] to the Cantor set C is discontinuous at every point in [0,1] of the form 1/2ⁿ. In addition, show that g is continuous at all other points in [0,1]. (See the graph of this function on the weekly summary page.) Hint: Note that g has a jump (of size 1/3ⁿ) only at the points that are of the form 1/2ⁿ.

Recall the definitions;

A function g is continuous at the point x if for any e greater than zero (think of e as being very small) there exists a d such that if |x-y| is less than d, then |g(x)-g(y)| is less than e (that is, if a point y is close enough to x then the value of the function g(y) at y is closer than e to the value of the function g(x) at x).

A function g is discontinuous at the point x if there exists an e greater than 0 such that for any d greater than 0, there is a point y such that |x-y| is less than d and |g(x)-g(y)| is greater than e (this is the negation of the definition of g being continuous at x).

Prove, using ternary expansions, that between every point in the Cantor set is a point that is not in the Cantor set. (Thus, this is another proof that the Cantor set contains no intervals; that it is 'dust'.)

For questions that require the use of binary or ternary expansions, you may want to review some properties of those expansions given in the formula page.

Show that every point in the Cantor set C is an accumulation point of points from the set E of end points of intervals removed during the construction of C (E = {0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, ...}). That is, show that for any p in C and any d>0 (small number), there is a q in E such that |p-q|< d. [This tells us that the end points "cluster" around the points in the Cantor set.]
(Hints: Note that if q in E, then the ternary expansion of q ends in infinitely many zeros and also contains no 1's. And that if x and y are two numbers in [0,1] whose ternary expansions agree to the nth place, then |x-y|<1/(3ⁿ); see the formula page for an explaination of this. )

Show that the 'Cantor middle-fifths' set has the same properties as the Cantor middle-thirds set (the one we discussed in class). So show that the Cantor middle-fifths set (1) has length 0, (2) has at least as many points as the interval [0,1], (3) is totally disconnected - is "dust" (i.e., contains no intervals), and (4) is self-similar. The proofs of these assertions should be similar to the proofs in class for the Cantor middle-thirds set, and at the same level of mathematical rigour.
(The construction of the Cantor middle-fifth set removes the middle fifth open interval of each subinterval that occurs at that stage. Note that in the n^th step 4^n-1 intervals are removed; in contrast, only 2^n-1 intervals were removed in the n^th step of the construction of the Cantor middle-thirds set. If you are not sure how the Cantor middle-fifths set is constructed, ask!)
(Hint: For (2), consider [x]₄ and [x]₅.)

Using an appropriate system of addresses, demonstrate the self-similarity of the square von Koch curve (as we did in class for the von Koch curve). Here is a picture of the von Koch curves for reference.

Compute the exponent d for the square von Koch curve. (It should be larger than the exponent for the (triangle) von Koch curve; this indicates that the square von Koch curve is more "complicated" than the von Koch curve).

Let K₁₀ be the curve obtained in the 10^th stage of the von Koch curve construction. Show that the plot of log(u) vs log(1/s) for K₁₀ is composed of two straight lines, one having positive slope and one having slope zero (use s_n = (1/3)ⁿ ). The line of slope zero extends arbitrarily far to the right ( i.e., towards smaller scales). What is the slope of the straight line segment of positive slope? At what value of s does the slope change?