Metric Number Theory: the good and the bad
Rebecca Thorn · Queen Mary Research Online (Queen Mary University of London) · 2005
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.Irrational numbers are also associated with the notion of 'I/J-well approximable.Let 1/J : R+ -+ R+ be a real, positive, decreasing function and consider the setHere and throughout this text 'Lm.' denotes infinitely many and the above function 1/J will be refered to as an approximation function.An element of the set W{1/J) is said to be a 1/J-well approximable number.Returning to the set of badly approximable numbers, it is easily verified that Bad C IR \ W(1/Jo) .1i£(F) := inf { ~ J(r,) : {B,} is a pcover of F} where the infimum is over all p-covers.Then the Hausdorff I-measure,