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.

In this paper it is shown that if the volume sum ∑r = 1∞ Ψ(r) converges for a monotonic function Ψ then the set of points (x, z, w) ∈ ℝ × ℂ × ℚp which simultaneously satisfy the inequalities |P(x)| ≤ H−v1 Ψλ1(H), |P(z)| ≤ H−v2 Ψλ2(H) and |P(w)|p ≤ H−v3 Ψλ3(H) with v1 + 2v2 + v3 = n − 3 and λ1 + 2λ2 + λ3 = 1 for infinitely many integer polynomials P has measure zero.

In this paper, we show that if the sum ∞ r=1 Ψ (r) diverges, then the set of points (x, z, w) ∈ R × C × Q p satisfying the inequalities |P (x)| < H −v1 Ψ λ1 (H), |P (z)| < H −v2 Ψ λ2 (H), and |P (w)| p < H −v3 Ψ λ3 (H) for infinitely many integer polynomials P has full measure. With a special choice of parameters v i and λ i , i = 1, 2, 3, we can obtain all the theorems in the metric theory of transcendental numbers which were known in the real, complex, or p-adic fields separately.

Abstract. Let p be a prime number. For a positive integer n and a p-adic number ξ, let λ n (ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ||qξ|| p , ||qξ 2 || p , . . . , ||qξ n || p are all less than q −λ−1 . Here, ||x|| p denotes the infimum of |x − n| p as n runs through the integers. We study the set of values taken by the function λ n .

No abstract