tailieunhanh - Đề tài " A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures "

A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of Rk to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former. . | Annals of Mathematics A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures By VictorBeresnevichn andSanju Velanin Annals of Mathematics 164 2006 971 992 A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures By Victor Beresnevich and Sanju Velani Dedicated to Tatiana Beresnevich Abstract A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of Rk to Hausdorff measure theoretic statements. In view of this the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former. 1. Introduction Throughout p R R will denote a real positive function and will be referred to as an approximating function. Given an approximating function p a point y y1 . yk 6 Rk is called simultaneously p-approximable if there are infinitely many q 6 N and p p1 . pk 6 Zk such that 1 y _ p ppf Pi q 1 1 . k . The set of simultaneously p-approximable points in Ik 0 1 k will be denoted by Sk p . For convenience we work within the unit cube Ik rather than Rk it makes full measure results easier to state and avoids ambiguity. In fact this is not at all restrictive as the set of simultaneously p-approximable points is invariant under translations by integer vectors. The pairwise co-primeness condition imposed in the above definition clearly ensures that the rational points p1 q . pk q are distinct. To some extent Research supported by EPSRC GR R90727 01. Royal Society University Research Fellow 972 VICTOR BERESNEVICH AND SANJU VELANI the approximation of points in Ik by distinct rational points should be the main feature when defining Sk 0 in which case .