It is mathematically impossible to pack this specific type of infinite space with perfect balls without leaving any gaps.
April 17, 2026
Original Paper
Tilings and coverings by balls in $\ell_1$
arXiv · 2604.15092
The Takeaway
Since 1981, mathematicians have wondered if you could perfectly 'tile' an infinite-dimensional space called ell-one with balls. Think of it like trying to fill a room with beach balls so perfectly that there isn't a single microscopic air gap left. This paper finally proves it is a lost cause—the geometry of this space just won't allow it. It settles a 40-year-old mystery about how the most abstract shapes in the universe fit together. For us, it defines the ultimate mathematical limits of how information can be packed in complex systems.
From the abstract
A famous result of Klee from 1981 is that the Banach space $\ell_1(\kappa)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $\kappa$ with $\kappa^\omega =\kappa$. Klee also observed that the smallest cardinal in which such a tiling might exist is $\kappa= 2^{\aleph_0}$, leaving open the question whether, for $\kappa< 2^{\aleph_0}$, $\ell_1(\kappa)$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that $\ell_1$ d