Systems of linear forms and covers for star bodies

In this paper Diophantine approximation involving general distance and error functions will be considered for systems of linear forms. Distance functions will be denoted by F : R → R and are continuous, non-negative and satisfy F (ux) = uF (x) for all u ≥ 0 ([3], p. 103). An error function ψ : R \ {0} → R is positive and satisfies ψ(x) → 0 as |x| → ∞, where |x| = max{|x1|, . . . , |xn|}. An upper bound depending on a cover for a star body associated with the distance function F (see [8]) and a restriction ψ1 of the error function ψ for the Hausdorff dimension of sets of linear forms (regarded as points in R) is obtained. The Hausdorff dimension is also expressed in terms of the Hausdorff dimension of sets of vectors (in R) and of the lower order (defined below) of the error function ψ1 (corresponding to the case m = 1) when the functions ψ and ψ1 satisfy suitable decay conditions. This combines and extends the results of [4], [5] and [13]. Definitions and properties of Hausdorff dimension are given in [7]. First some notation and definitions are needed. Throughout m,n and N will be positive integers and q ∈ Z. The system a1x1j + . . .+ amxmj , 1 ≤ j ≤ n , of n real linear forms in m real variables a1, . . . , am will be written more concisely as aX where a ∈ R and where X is anm×n real matrix. The set of real m× n matrices will be identified with R in the usual way (i.e., by identifying the matrix X = (xij) with the point (x11, . . . , x1n, x21, . . . , xmn) in R). For each x in R, denote the symmetrised fractional part of x by 〈x〉 = x − kx , where kx is the unique integer vector such that x − kx ∈ (− 1 2 , 1 2 ] . Thus |〈x〉| = max{‖x1‖, . . . , ‖xn‖} ,