Diophantine exponents for mildly restricted approximation

Abstract

We are studying the Diophantine exponent μ_{n,l}$ defined for integers 1 \leq l < n and a vector α\in \mathbb{R}^n by letting μ_{n,l} = \sup{μ\geq 0: 0 < ||x \cdot α|| < H(x)^{-μ} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,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, \infty), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors αwith μ_{n,l} (α) = μfor μ\geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors αfor which the gaps in the increasing sequence μ_{n,1} (α) \leq ... \leq μ_{n,n-1} (α) \leq w_n (α) can be chosen to be arbitrary.

20 pages