[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 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