Top new questions this week:

Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$?
What I know:
...

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...

Euler proved
$$\frac{\pi}{6}=1\frac{1}{2}\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $$, prime ...

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less welldocumented than the main narrative of ...

Consider, on the one hand:
the CurryHoward correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$calculus, and on the other, propositions and ...

Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows:
For two ultrafilters $U,W$ on $\omega$, we say ...

Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$Hölder continuous  that is, it is not $\alpha$Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...

Greatest hits from previous weeks:

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward.
Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...

Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John:
If you were to come back a hundred years after your death, what problem ...

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in
the game's structure,
optimal strategies,
practical strategies,
analysis of the game ...

As a nonnative English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main result(...

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for nontrivial results?
(One could ask if this is of interest to mathematicians, and I ...

Can you answer these questions?

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some BoreltoBorel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...

The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...

One can motivate Roth's theorem as follows. On $[0,1]$ consider the function $f$ that takes
$x$ to the cardinality of the set
{ $p/q : x  p/q  < 1/q^{2 + \epsilon}$ } .
Now one can see that $\...
