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Using Fourier analysis in the $d$ variable, we can get the optimal upper bound. As in Tao's argument, if there is a distribution of sequences for which each pair is $r$-probably increasing, there mus …

$\Sigma$ is clearly not a measurable set in the product sigma-algebra, moreover it is so non-measurable that every measurable set containing it is the whole set (any any measurable set contained in it …

I can prove that the number actually described by the word problem, which is $$ \prod_{i=0}^{\infty} \left( 1- \frac{1}{2^{2^i}} \right),$$ is irrational, by a method similar to David Speyer's. Expand …

I'm going to give an answer that discusses some things that the other answers don't go into as much detail on. In particular let me try to explain why the results you mention on classical groups havin …

In dimension $5$ and above, a random angel escapes a blind devil with positive probability, as long as $r$ is sufficiently large. To see this, let's replace the angel with one that chooses a point to …

Calculus of variation shows that the segment of the boundary of the optimal shape where the tangent line intersects two opposite sides must be a parabola. To see, this observe that the tangent line …

Let $F(x) =\int_{1-x}^{\infty} W(t) dt$. Then the inequality is $F(\rho x ) \geq \alpha F(x)$ and we also have that $\frac{dF}{dx} (\frac{1}{2}) = 1$ and $F$ is convex down. In particular for $x=\frac …

This doesn't really require modern functional analytic tools, but we can prove a statement (due originally to Edelheit, according to Jochen Wengenroth in the comments) like Let $V$ be a Frechet space …

No. If $\mathbb E[X^+]=\infty$ then $$\infty = \mathbb E[ \lfloor X^+\rfloor ] =\sum_{n=1}^{\infty} \mathbb P(X \geq n) = \sum_{n=1}^{\infty} \mathbb P(X_n\geq n)$$ and the events $X_n \geq n$ are ind …

Aaron and fedja have pointed out that the problem is equivalent to finding the convex region in the plane with area $1/2$ with the highest probability that a random line does not intersect it. The o …

One way to see this technique is as a way of dealing with certain bad cases. $E[P/Q]$ can be unhelpfully dragged up by the inclusion of certain cases where $Q$ is small and $P$ is medium. $E[P]/E[Q]$ …

I can show that $N(\epsilon)$ is equal to $\epsilon^{-2}$ up to a log factor on each side. The strategy I'll use is to give an upper bound for $\pi(1/2+\epsilon,n)$. Optimizing it, we obtain an upper …

There's no particular reason to use a continuous space-filling curve, other than style points. Just expand x in binary and send the odd digits to $f$ and the even digits to $g$. To prove that you get …

Yes, and it's proportional to $\cos ( \pi/p)^n$. To see this, observe that adding one more independent random variable acts on the probability distribution as a linear operator, hence a $p \times p$ m …

Let $(Y_1,\dots, Y_k)$ be a random $k$-tuple uniformly distributed in $(\mathbb R/\mathbb Z)^k$. Let $X_i = \sum_{j=1}^k i^j Y_j \in \mathbb R/\mathbb Z$. Then the $X_i$ are $k$-wise independent (and …