Sphere Packings in Euclidean Space with Forbidden Distances

In this paper, we study the sphere packing problem in Euclidean space, where we impose additional constraints on the separations of the center points. We prove that any sphere packing in dimension $48$, with spheres of radii $r$, such that \emph{no} two centers $x_1$ and $x_2$ satisfy $\sqrt{\tfrac{4}{3}} < \frac{1}{2r}|x_1-x_2| <\sqrt{\tfrac{5}{3}}$, has density less or equal than $( 3\pi /2)^{24}/24!$. Equality occurs if and only if the packing is given by a $48$-dimensional even unimodular extremal lattice. This shows that any of the lattices $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ are optimal for this constrained packing problem. We also give results for packings up to dimension $d\leq 1200$, where we impose constraints on the distance between centers and on the minimal norm of the spectrum, and show that even unimodular extremal lattices are again uniquely optimal. Moreover, in the $1$-dimensional case, we give a condition on the set of constraints that allow the existence of an optimal periodic packing, and we develop an algorithm to find them by relating the problem to a question about linear domino tilings.Comment: 39 page

Monsky’s Theorem / O Teorema de Monsky

The main objective of this paper is to prove Monsky’s Theorem, that provides a beautiful application of the 2-adic valuation in order to solve a plane geometry problem. This theorem states that given any dissection of a square into finitely many nonoverlapping triangles of equal area the number of triangles must be even. In order to prove this statement, we will need some previous results from Combinatorial Topology and Algebra.