Journal of the London Mathematical Society, Vol. 20 (1945), 22-26

[Received 15 June 1945]

Introduction

Very recently I proved the theorem:

Theorem 1.If x = αξ+βη, y = γξ+δη are two (real) binary linear forms of positive determinantΔ = 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* F_{μ} 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 x_{r}= Σ_{(s=1..n)}a_{rs}ξ_{s}(r = 1, 2, ..., n) are n (real) linear forms of positive determinantΔ = |a

_{rs}|,and if μ is any given number in the open interval 0 < μ < 1, then integers ξ

_{s}can be found such thatμm

^{n}√Δ<max(|x_{1}|, |x_{2}|, ..., |x_{n}|)<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