On sparse geometry of numbers

TL;DR

This paper explores the sparse geometric structure of lattices, providing bounds and conditions for sparse vectors, and connects these properties to algebraic and geometric structures like elliptic curves and modular curves.

Contribution

It introduces bounds on sparse vectors in lattices, establishes conditions for virtual rectangularity, and links lattice properties to elliptic curves and modular curves.

Findings

Bounds on the sup-norms of sparse vectors in lattices

Conditions for lattices to be virtually rectangular

Characterization of planar virtually rectangular lattices via rationality conditions

Abstract

Let $L$ be a lattice of full rank in $n$-dimensional real space. A vector in $L$ is called $i$-sparse if it has no more than $i$ nonzero coordinates. We define the $i$-th successive sparsity level of $L$, $s_i(L)$, to be the minimal $s$ so that $L$ has $s$ linearly independent $i$-sparse vectors, then $s_i(L) \leq n$ for each $1 \leq i \leq n$. We investigate sufficient conditions for $s_i(L)$ to be smaller than $n$ and obtain explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in~$L$. This result can be viewed as a partial sparse analogue of Minkowski's successive minima theorem. We then use this result to study virtually rectangular lattices, establishing conditions for the lattice to be virtually rectangular and determining the index of a rectangular sublattice. We further investigate the $2$-dimensional situation, showing that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real $j$-invariant. We also identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points.