515 – Homework 1

This set is due Feb. 8 at the beginning of lecture. Of course, let me know if more time is needed or anything like that.

0. During lecture I have sometimes skipped some arguments or not given as much detail as you may have wanted. If there was a result that in particular required of you some effort to complete in detail, please state it here and show me how you filled in the gaps left in lecture. Also, if there is a result for which you do not see how to fill in the details, let me know as well, as I may have overlooked something and it may be worth going back over it in class.

1. Give an example of a bounded set $E\subset{\mathbb R}$ for which $\displaystyle \lim_{N\to\infty}\frac1N|E\cap\frac1N{\mathbb Z}|$

does not exist.

2. Compute $m^{*,(J)}({\mathbb Q}\cap[0,1])$.

3. From the book, solve exercises 1.1.3, 1.1.5, 1.1.6, and 1.1.15.

[To get you started on 1.1.3: First verify in ${\mathbb R}$ that $m'$ assigns value 0 to any point. For this, use monotonicity and translation invariance, arguing first that $m'((0,1/n))\le m'([0,1])/n$ for any $n$. Then find that $m'([0,1/k)^n)$ in terms of $m'([0,1]^n)$, and use this to find $m'(B)$ for any box with rational coordinates. Use this to compute $m'(B)$ for any box, and conclude by analyzing arbitrary elementary sets.

Note we essentially solved 1.1.15 in class, but under the assumption that 1.1.6 holds.]

4. From the book, solve Exercises 1.1.7-10. Make sure to explain in 1.1.9 why Tao’s definition of compact convex polytopes coincides with what should be our intuitive definition. Please also verify that convex polytopes are indeed convex.

(For a nice argument verifying that indeed $c_d=\frac1d\omega_d$, at least for even values of $d$, see the paper “On the volumes of balls” by Blass and Schanuel, available here.)

5. From the book, solve exercise 1.1.11.

(If you are not comfortable with linear algebra beyond size $3\times 3$, at least argue in the plane and in ${\mathbb R}^3$.)

6. From the book, solve exercise 1.1.13.

7. From the book, solve exercise 1.1.17.

One Response to 515 – Homework 1

1. Andres Caicedo says:

(I have added a missing prime in the hint on question 3.) Thanks to Tara for noticing it.