Diophantine exponents for mildly restricted approximation

• Published in: Arkiv för matematik Vol. 47; no. 2; pp. 243 - 266
• Main Authors: Bugeaud, Yann, Kristensen, Simon
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2009
• Subjects: Mathematics; Mathematics and Statistics; Algebraic Number; Minimal Polynomial; Diophantine Approximation; Cantelli Lemma; Acta Arith
• ISSN: 0004-2080; 1871-2487
• DOI: 10.1007/s11512-008-0074-0

We are studying the Diophantine exponent μ n , l defined for integers 1≤ l < n and a vector α∈ℝ n by letting where is the scalar product, denotes the distance to the nearest integer and 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 μ n , l (α)=μ for μ≥ n . Finally, letting w n denote the exponent obtained by removing the restrictions on , we show that there are vectors α for which the gaps in the increasing sequence μ n ,1 (α)≤...≤μ n , n -1 (α)≤ w n (α) can be chosen to be arbitrary.