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I'm working through Measure and Category by Oxtoby with some friends, but since the book doesn't have any exercises, we needed to come up with our own to discuss. My friend found the following problems on the Baire Category theorem (from Rudin and Sally), which were a good difficulty level for us.

  1. Prove that $\mathbb{Q}$ is not a countable intersection of open sets.
  2. Let $ \{ f_n\} $ be a sequence of continuous functions on $\mathbb{R}$ so that for every $x$, $\lim_{n \to \infty}f_n(x)$ exists and is finite. Prove that there is an interval $(a, b)$ of positive length so that the set $\{|f_n(x)| : n \in \mathbb{N}, x \in (a,b)\}$ is bounded above.
  3. Does there exist a sequence of continuous positive functions $f_n$ on $\mathbb{R}$ so that the set $\{f_n(x)\}$ is unbounded if and only if $x$ is rational? What if rational is replaced with irrational?
  4. Prove that if $ \{ f_n\} $ is a sequence of continuous functions from $ \mathbb{R}$ to $ \mathbb{R} $ so that for each $x$, $ f(x) = \lim_{n \to \infty}f_n(x)$ exists and is finite, then for each $\varepsilon > 0 $ there is a nonempty open set $U $ and a large $N$ so that $ |f(x) - f_n(x)| < \varepsilon$ for all $n \geq N$, $x \in U $.
  5. Prove that if $f$ is a continuous function from positive reals to positive reals so that $f(x), f(2x), f(3x), \ldots $ tends to $0$ for all $x$, then $f(t) \to 0$ as $t \to \infty$.

It would be greatly appreciated if someone could provide a reference to some problems of similar difficulty involving Liouville numbers.

  • You want questions on measure and category involving Liouville numbers? or just any old questions involving Liouville numbers, whether they have to do with measure/category or not? – Gerry Myerson Dec 16 '20 at 02:38
  • @GerryMyerson hopefully ones involving measure, but I'm not sure how many of those there are, so others would do as well. –  Dec 16 '20 at 02:44
  • I have a couple of lengthy lists of problems (.pdf files -- one 40 pages, the other 27 pages) I wrote in the early 2000s for Ph.D. qualifying exam studying purposes for students where I was then teaching at, many with hints and even solution outlines along with pointers to the literature for additional information and related aspects of the problem, and quite a few of those problems would qualify for what you want. If you send me an email (mine is "given" at my profile), I can reply with those problem lists. – Dave L. Renfro Dec 17 '20 at 07:49

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Here are a couple of results about Liouville numbers that you can take for exercises, if you like. Both are from Burger and Tubbs, Making Transcendence Transparent.

Theorem 1.6. Every real number can be expressed as a sum of two Liouville numbers.

Corollary 6.24. Let $\alpha$ be a nonzero algebraic number, and $L$ a Liouville number. Then $e^{\alpha}+L$ is transcendental.

The second is considerably harder than the first (but I don't see why $\alpha$ has to be nonzero).

Erdos proved that every nonzero real number can be written as a product of two Liouville numbers. I don't know how hard this is.

Gerry Myerson
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