Counting basis extensions in a lattice

TL;DR

This paper investigates the asymptotic count of basis extensions from primitive collections in integer lattices, connecting to unimodular matrices, Farey fractions, and multilinear forms, with results applicable to general lattices.

Contribution

It provides a new asymptotic estimate for extending primitive collections to bases with bounded vectors, generalizes to arbitrary lattices, and explores properties of sparse multilinear representations.

Findings

Asymptotic estimate for basis extensions with bounded vectors

Counting lattice points in hyperplanes with sup-norm constraints

Properties of sparse integer representations via multilinear forms

Abstract

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the $2$-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in~$\mathbb R^n$. Our main result on counting basis extensions also generalizes to arbitrary lattices in~$\mathbb R^n$. Finally, we establish some basic properties of sparse representations of integers by multilinear forms.