Lattice tilings of Hilbert spaces
Carlo Alberto De Bernardi, Tommaso Russo, Jacopo Somaglia
TL;DR
The paper resolves Fonf and Lindenstrauss's question by constructing a lattice tiling of a non-separable Hilbert space $\ell_2(\Gamma)$ using translates of a bounded symmetric convex body, yielding an equivalent norm with a ball tiling. The core method reduces the problem to finding a $2^{1/p}$-separated, $1$-dense subgroup of $\ell_p(\Gamma)$, enabling Voronoi tilings that are lattice for $p=2$ and extend to $p\in(1,\infty)$, with a direct ball tiling for $p=1$. It further shows that lattice tilings by balls are not disjoint and analyzes the intersection structure, while also proving that every infinite-dimensional normed space contains a $1$-separated, $(1+\varepsilon)$-dense subgroup with generators of norm at most $2+\varepsilon$, providing a simpler proof that discrete subgroups are free. The work also poses open questions on separable tilings, point-finite tilings, and the existence of rotund or smooth ball tilings, guiding future exploration in infinite-dimensional tiling theory.
Abstract
We construct a bounded and symmetric convex body in $\ell_2(Γ)$ (for certain cardinals $Γ$) whose translates yield a tiling of $\ell_2(Γ)$. 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 $\ell_1(Γ)$ yields a point-$2$-finite lattice tiling by balls of radius $1$ for $\ell_1(Γ)$, 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 Zsák. We also give an alternative elementary proof of Steprāns' result that discrete subgroups of normed spaces are free.