Sets of exact 'logarithmic' order in the theory of Diophantine approximation

Abstract

For each real number α, let E(α) denote the set of real numbers with exact order α. A theorem of Güting states that for α ≥ 2 the Hausdorff dimension of E(α) is equal to 2/α. In this note we introduce the notion of exact t-logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Background and statement of results The classical theoryFor each real number τ , let W (τ ) denote the set of real numbers which are τ -well approximable, that is W (τ ) := {x ∈ R : |x − p/q| ≤ |q| −τ for infinitely many rationals p/q} . For a real number x, its exact order τ (x) is defined as follows:τ (x) := sup{τ : x ∈ W (τ )} .It follows from Dirichlet's theorem in the theory of Diophantine approximation that τ (x) ≥ 2 for all x ∈ R. For α ≥ 2, let E(α) denote the set of numbers with exact order α; that is E(α) := {x ∈ R : τ (x) = α} . Royal Society University Research Fellow 254 V. Beresnevich et al.The set E(α) is equivalent to the set of real numbers x for which Mahler's function θ 1 (x) is equal to α − 1; the general function θ n (x) is central to Mahler's classification of transcendental numbers (see [1,4]). In [4], Güting proved the following result:Throughout, dim X will denote the Hausdorff dimension of the set X. For further details regarding Hausdorff dimension and measure see Sect. 1.3. At this point it is worth mentioning the classical result of Jarník and Besicovitch which states that dim W (α) = 2/α. Thus, Güting's result says that the 'size' of the sets E(α) and W (α) expressed in terms of Hausdorff dimension are the same. In this note we give a short proof of Güting's theorem and show that the s-dimensional Hausdorff measure of the set E(α) at the critical exponent is infinity, that isThese exact order results are a simple consequence of our results on exact tlogarithmic order. Linear forms and exact t-logarithmic orderLet ψ : R + → R + be a real positive function. An m×n matrix X = (x ij ) ∈ R mn is said to be ψ-well approximable if the system of inequalitiesis satisfied for infinitely many vectors q ∈ Z m , p ∈ Z n . Here |q| denotes the supremum norm of the vector q ; i.e. |q| = max{|q 1 | , . . . , |q m |} . The systemof n real linear forms in m variables q 1 , . . . , q m will be written more concisely as q X ,where the matrix X is regarded as a point in R mn . In view of this notation, the set of ψ-well approximable points will be denoted bywhere 'i.m.' means 'infinitely many'. By definition, |qX − p| = max 1≤j ≤n |q.X (j ) − p j | where X (j ) is the j'th column vector of X. In the case when ψ(r) = r −τ we denote W (m, n; ψ) by W (m, n; τ ) and note that when m = n = 1 the ...

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