ON A QUESTION OF MORDELL AND A SPECTRUM OF LINEAR FORMS

Abstract

Let L₁(x), L₂(x), …, L_N(x) be N real linear forms in N variables defined by

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such that the associated N × N matrix of coefficients C = (c_mn) 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, …, ε_N satisfying ε₁, ε₂, …, ε_N [ges ] 1, there exists a lattice point p ∈ ℤ^N, 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 Δ_N such that for each N × N unimodular real matrix C, there exist positive real numbers ε₁, ε₂, …, ε_N satisfying

formula here

such that the only integer solution to the inequalities of (1.1) is p = 0? Moreover, if Δ_N does exist, what is the best possible value for Δ_N, that is, what is the supremum of all such admissible values of Δ_N? Clearly, by Minkowski's theorem, if Δ_N exists, then Δ_N < 1.