## Kathleen Ollerenshaw

### Lattice Points in a Hollow n-Dimensional Hypercube Journal of the London Mathematical Society, Vol. 20 (1945), 22-26

Introduction

Very recently I proved the theorem:

Theorem 1. If x = αξ+βη, y = γξ+δη are two (real) binary linear forms of positive determinant

Δ = αδ–βγ

and if μ is any given number in the open interval 0 < μ < 1, then integers ξ, η can be found such that

μm √Δ < max(|x|, |y|) < m √Δ,

where m is the greatest integer in 1/(1–μ).

This result was obtained by considering the lattices of minimum determinant which have no point inside the square frame Fμ any point (x,y) of which satisfies the inequalities

μ < max(|x|,|y|) < 1   (0 < μ < 1).

Such lattices do in fact exist; they are said to be "critical". In this note I prove the more general theorem:

Theorem 2. If xr = Σ(s=1..n) arsξs (r = 1, 2, ..., n) are n (real) linear forms of positive determinant

Δ = |ars|,

and if μ is any given number in the open interval 0 < μ < 1, then integers ξs can be found such that

μm n√Δ < max(|x1|, |x2|, ..., |xn|) < m n√Δ,       (3)

where m is the greatest integer in 1/(1–μ).

This is a "best possible" result in the sense that, if μ and Δ are given, then m in (3) is the least number independent of the a's for which this assertion is true for all linear forms of determinant Δ