Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sizes of sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set $D$ of positive integers, call a set $S$ "$D$-avoiding" if no two elements of $S$ differ by an element of $D$. It is shown that any $D… Expand

We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent… Expand

The notion of elementary numerosity as a special function dened on all subsets of a given set which takes values in a suitable non-Archimedean field, and satises the same formal properties of finite cardinality is introduced.Expand

The notion of a completely saturated packing [Fejes Toth, Kuperberg and Kuperberg,
Highly saturated packings and reduced coverings, Monats. Math. 125 (1998) 127-145] is a
sharper version of maximum… Expand

It is proved that for all but countably many radii, optimally dense packings must have low symmetry in the hyperbolic space of any dimension m≥ 2.Expand

This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller,… Expand