Let L1(x), L2(x), …,
LN(x) be N real linear forms in N variables defined by
formula here
such that the associated N × N matrix of coefficients
C = (cmn) is unimodular. In the
classical theory of geometry of numbers, Minkowski's well-known linear forms
theorem asserts that for any positive numbers ε1, ε2, …, εN
satisfying ε1, ε2, …, εN [ges ] 1,
there exists a lattice point p ∈ ℤN, p ≠ 0, such that
formula here
In 1937, Mordell [7] posed the following question which may be viewed, in some
sense, as a converse to Minkowski's result. Does there exist a constant ΔN such
that for each N × N unimodular real matrix C, there exist positive real numbers
ε1, ε2, …, εN satisfying
formula here
such that the only integer solution to the inequalities of (1.1) is p = 0? Moreover, if
ΔN does exist, what is the best possible value for
ΔN, that is, what is the supremum
of all such admissible values of ΔN? Clearly, by Minkowski's theorem, if ΔN exists,
then ΔN < 1.