Counting basis extensions in a lattice

Forst, Maxwell; Fukshansky, Lenny

doi:10.1090/proc/16011

Counting basis extensions in a lattice
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by Maxwell Forst and Lenny Fukshansky

Proc. Amer. Math. Soc. 150 (2022), 3199-3213

DOI: https://doi.org/10.1090/proc/16011

Published electronically: May 6, 2022

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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.

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