[Received 20 December 1944]
A problem of interest in the theory of numbers is to determine the lower bound of |f(x,y)| for integer values of x,y, where f is a real function of x,y. Minkowski showed that, if the region R defined by
(1) | f(x,y) | < 1
is closed, convex, and symmetrical about the origin, and if Λ is a point lattice defined by
x = αξ+βη, y = γξ+δη, Δ = αδ–βγ > 0,
where ξ, η run through all integers not both zero, then the problem is equivalent to that of finding the minimum value (here actually attained) of Δ, say Δ(R), extended over all admissible lattices Λ, i.e. lattices which have no point inside R. If an admissible lattice Λ can be found whose determinant is actually Δ(R), we call it a critical lattice of the region R. Every critical lattice gives rise to a form f(x,y) for which the equality sign is necessary in (1), and we therefore seek to determine the critical lattices of R as well as the value of Δ(R). We extend the definitions of admissible and critical lattices to include domains of any type which are symmetrical about the origin.
Recently Mordell and Mahler have considered (by independent methods) a number of infinite and non-convex "star" domains, that is domains which are symmetrical about the origin and in which every radius from the origin meets the boundary in only one point. In this note (at Dr. Mahler's suggestion) I consider a hollow or ring-shaped region in which, of course, the latter condition no longer holds. The simplest example of such a region is the square frame (Fμ say) defined by the inequalities
μ < max(|x|,|y|) < 1 (0 < μ < 1).
I find that Δ(Fμ) is a simple discontinuous function of the parameter μ. Mahler has shown that, for a star domain Kμ (say) such that Kμ contains Kν if μ > ν, Δ(Kμ) is a never decreasing continuous function of μ. The region Fμ gives the first known example of a discontinuous result. This is probably characteristic of hollow, ring-shaped regions.