Geometry of Continued Fractions — Faces of sails

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integer $n\ge1$, a lattice $n$-simplex is the convex hull $P=\operatorname{conv}\{v_0,\ldots,v_n\}$ of $n+1$ affinely independent points in $\mathbb Z^n$. It is empty when \[P\cap\mathbb Z^n=\{v_0,\ldots,v_n\};\] thus extra lattice points on the boundary, as well as in the interior, are excluded. Two such simplices are integer congruent if $P'=UP+a$ for some $U\in\operatorname{GL}_n(\mathbb Z)$ and $a\in\mathbb Z^n$.

Statement recorded in the supplied source.

For every dimension $n\ge1$, classify all empty lattice $n$-simplices up to integer congruence. [S2]

Short English statement

Classify lattice simplices that contain no lattice points except their vertices, allowing every dimension and identifying simplices related by an invertible integer affine change of coordinates.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1900004 (AMR-018-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Oleg Karpenkov, Open problems in geometry of continued fractions, 2017. Problem 4 and its surrounding definitions and remarks. https://arxiv.org/pdf/1712.01450.