Simultaneous Diophantine Approximation on Polynomial Curves

Abstract

Abstract. The Hausdorff dimension and measure of the set of simultaneously ψ-approximable points lying on integer polynomial curves is obtained for sufficiently small error functions. §1. Introduction and notation. In dimensions higher than one there are two standard forms of Diophantine approximation and they have rather different properties. To describe these ideas some notation and terminology is needed. For each t ∈ R letThe supremum norm will be denoted by | . |, that is, for a vector x ∈ Z n , |x| = max{|x 1 |, . . . , |x n |}.Throughout this paper, hcf(x, y) will be used to denote the highest common factor of the integers x and y.Let ψ be a decreasing function such that ψ(r ) → 0 as r → ∞. The set S ψ (M) of simultaneously ψ-approximable points lying on an m-dimensional manifold M embedded in R n is defined byThere is a natural dual to this set, namelyIf ψ(r ) = r −τ then the sets are denoted S τ (M) and L τ (M), respectively. Obviously any element of Q n lying on M is in S ψ (M) for all functions ψ. Correspondingly, the intersection of M with a rational hyperplane given by the equation q.x = p (for p ∈ Z and q ∈ Z m ) is contained in L ψ (M) for all ψ. Any other points in either S ψ (M) or L ψ (M) lie "close" to these points or planes. In this paper we will study S ψ (M) when M is a polynomial curve and ψ(r ) → 0 sufficiently fast.

On simultaneous rational approximation to a real number and its integral powers

“…This was already established in [5], but with a different proof. It follows from Theorem 3.7 that, for any n 2, the spectrum of λ n includes the interval [n − 1, +∞], a weaker conclusion than Theorem 3.4.…”

“…-We use Theorem 1 of [5] to construct a suitable dimension function f such that the Hausdorff f -measure of the set defined in (4.1) is positive, while, for every positive integer h and for i = 1, . .…”

Intrinsic Diophantine Approximation on General Polynomial Surfaces

“…In this paper, we aim to establish a Jarník‐type zero–infinity law for the Hausdorff measure of . In the case where , this has been studied previously by Budarina et al in [5]. The special case of Veronese manifolds is covered in [9, §2].…”

Hausdorff Dimension and Diophantine Approximation

“…The Jarník-Besicovitch theorem (1.3) was recently extended by Budarina, Dickinson, and Levesley [9] as follows (see also [12] for an alternative proof).…”