Let $E\subset \mathbb{R}^n$ be a cross-polytope: $$E= \left\lbrace x : \frac{|x_1|}{q_1}+\cdots+\frac{|x_n|}{q_n}\leq 1 \right\rbrace, $$ where $q_1,\dots,q_n$ are positive integer …

Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$. By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyper …

Let $X$ be a compact metrizable topological semilattice with a neighbourhood base of sub-semilattices that are path connected. I need a reference for a proof that $X$ is an absolut …

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 …

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda …

A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for …

I am wondering about generalisations of the concept of equivalence relations to suplattices. Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. …

Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector. Can thi …

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of typ …

I am not sure if its OK to ask this question here. Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a fu …

By  lattice  I'll mean  Birkhoff lattice. The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to b …

Let $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ be $n$ linearly independent vectors in an $n$-dimensional lattice $\Lambda$ in $\mathbb{R}^n$ and let $\mathbf{v}^*_1 ,\mathbf{v …

The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. …

Hi, overflowers. I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperpla …

Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$). Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w_v(M) …