Ubiquitous systems and metric number theory

• Published in: Advances in mathematics (New York. 1965) 2008-06, Vol.218 (2), p.368-394
• Main Author: Durand, Arnaud
• Format: Article
• Language: English
• Subjects: Diophantine approximation; Hausdorff dimension; Hausdorff measures; Heterogeneous ubiquity; Large intersection; Multifractal measures
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2007.12.008

We investigate the size and large intersection properties of E t = { x ∈ R d | ‖ x − k − x i ‖ < r i t for infinitely many ( i , k ) ∈ I μ , α × Z d } , where d ∈ N , t ⩾ 1 , I is a denumerable set, ( x i , r i ) i ∈ I is a family in [ 0 , 1 ] d × ( 0 , ∞ ) and I μ , α denotes the set of all i ∈ I such that the μ-mass of the ball with center x i and radius r i behaves as r i α for a given Borel measure μ and a given α > 0 . We establish that the set E t belongs to the class G h ( R d ) of sets with large intersection with respect to a certain gauge function h, provided that ( x i , r i ) i ∈ I is a heterogeneous ubiquitous system with respect to μ. In particular, E t has infinite Hausdorff g-measure for every gauge function g that increases faster than h in a neighborhood of zero. We also give several applications to metric number theory.