Diophantine Exponents for Mildly Restricted Approximation

Abstract. We are studying the Diophantine exponent μn,l defined for integers 1 ≤ l < n and a vector α ∈ R by letting μn,l = sup{μ ≥ 0 : 0 < ‖x · α‖ < H(x) for infinitely many x ∈ Cn,l ∩ Zn}, where · is the scalar product and ‖ · ‖ denotes the distance to the nearest integer and Cn,l is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l + 1,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μn,l(α) = μ for μ ≥ n. Finally, letting wn denote the exponent obtained by removing the restrictions on x, we show that there are vectors α for which the gaps in the increasing sequence μn,1(α) ≤ · · · ≤ μn,n−1(α) ≤ wn(α) can be chosen to be arbitrary.