Metric Number Theory: the good and the bad
MetadataShow full item record
Each aspect of this thesis is motivated by the recent paper of Beresnevich, Dickinson and Velani (BDV03]. Let 'ljJ be a real, positive, decreasing function i.e. an approximation function. Their paper considers a general lim sup set A( 'ljJ), within a compact metric measure space (0, d, m), consisting of points that sit in infinitely many balls each centred at an element ROt of a countable set and of radius 'I/J(130) where 130 is a 'weight' assigned to each ROt. The classical set of 'I/J-well approximable numbers is the basic example. For the set A('ljJ) , [BDV03] achieves m-measure and Hausdorff measure laws analogous to the classical theorems of Khintchine and Jarnik. Our first results obtain an application of these metric laws to the set of 'ljJ-well approximable numbers with restricted rationals, previously considered by Harman (Har88c]. Next, we consider a generalisation of the set of badly approximable numbers, Bad. For an approximation function p, a point x of a compact metric space is in a general set Bad(p) if, loosely speaking, x 'avoids' any ball centred at an element ROt of a countable set and of radius c p(I3Ot) for c = c(x) a constant. In view of Jarnik's 1928 result that dim Bad = 1, we aim to show the general set Bad(p) has maximal Hausdorff dimension. Finally, we extend the theory of (BDV03] by constructing a general lim sup set dependent on two approximation functions, A('ljJll'ljJ2)' We state a measure theorem for this set analogous to Khintchine's (1926a) theorem for the Lebesgue measure of the set of ('l/Jl, 1/12)-well approximable pairs in R2. We also remark on the set's Hausdorff dimension.
AuthorsThorn, Rebecca E
- Theses