Lattice tilings of Hilbert spaces

TL;DR

This paper constructs convex bodies in infinite-dimensional Hilbert spaces that tile the space via their translates, providing new examples of tilings in Banach spaces and exploring properties of discrete subgroups in normed spaces.

Contribution

It introduces the first known tilings of infinite-dimensional Hilbert spaces with convex bodies and analyzes their properties, also simplifying proofs related to discrete subgroups.

Findings

Constructed convex body tilings in $oldsymbol{ ext{ell}_2( ext{Gamma})}$ and $oldsymbol{ ext{ell}_1( ext{Gamma})$.

First example of an infinite-dimensional reflexive Banach space with a ball tiling.

Proved lattice tilings by balls are never disjoint and each tile intersects many others.

Abstract

We construct a bounded and symmetric convex body in $\ell_2(\Gamma)$ (for certain cardinals $\Gamma$) whose translates yield a tiling of $\ell_2(\Gamma)$. This answers a question due to Fonf and Lindenstrauss. As a consequence, we obtain the first example of an infinite-dimensional reflexive Banach space that admits a tiling with balls (of radius $1$). Further, our tiling has the property of being point-countable and lattice (in the sense that the set of translates forms a group). The same construction performed in $\ell_1(\Gamma)$ yields a point-$2$-finite lattice tiling by balls of radius $1$ for $\ell_1(\Gamma)$, which compares to a celebrated construction due to Klee. We also prove that lattice tilings by balls are never disjoint and, more generally, each tile intersects as many tiles as the cardinality of the tiling. Finally, we prove some results concerning discrete subgroups of normed spaces. By a simplification of the proof of our main result, we prove that every infinite-dimensional normed space contains a subgroup that is $1$-separated and $(1+\varepsilon)$-dense, for every $\varepsilon>0$; further, the subgroup admits a set of generators of norm at most $2+\varepsilon$. This solves a problem due to Swanepoel and yields a simpler proof of a result of Dilworth, Odell, Schlumprecht, and Zs\'ak. We also give an alternative elementary proof of Stepr\={a}ns' result that discrete subgroups of normed spaces are free.