Let $A \in M_n(\mathbb Z)$ and $\|A\| = \max |a_{ij}|$. Denote $$ S(r) = \sum_{\substack{\|A\| \leq r \\ \det{A} \neq 0}} \dfrac{1}{|\det{A}|} $$ - the sum over all matrices $A \in …

There are arbitrarily large finite sets of points in $\mathbb R^3$ whose Voronoi-domains intersect all pairwise in $2-$dimensional polytopes. This shows that one needs infinitely m …

In Weil's paper "On a certain type of characters of the idele-class group of an algebraic number field", Weil introduces a class of characters on the Idele class group (of not …

Dirichlet proved a classical theorem about approximating irrational real numbers with rational numbers, saying that for any irrational real number $\alpha$, you can find infinitely …

Hello everyone, I was doing some late night random reading and I got to wonder about some stuff about the Gauss circle problem. To begin with, consider a circle in $\mathbb{R}^{2} …

Given $A,B\in {\Bbb Z}^2$, write $A \leftrightarrows B$ if the interior of the line segment AB misses ${\Bbb Z}^2$. For $r>0$, define $S_r:=\{ \{A, B\} | A,B\in {\Bbb Z}^2,||A||& …

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real …

Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of l …

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face area …

Definition 1. Given a body $V$ in $\mathbb R^n$, the function $$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$ will be called profile of $V$. Definition 2. Define Voronoi cell of latti …

I'd like to state explicitly a problem which was somehow hidden in my three-week-old post: Does there exist an absolute constant $c>0$ with the property that for any matrix $M\ …

Let $n>1$ be an integer, and $a>1$ a real number. Consider the subspace $L<R^{2^n}$ generated by the $n$ possible tensor products of the $n-1$ copies of the vector $(1,a)$ and o …

Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector $x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by $\|\cdot\|$ both the Eucl …

I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and t …

In answer to Pete L. Clark's question http://mathoverflow.net/questions/39510/ on Euclidean quadratic forms, I gave an example in seven variables, repeated below. Pete's Euclidean …