## Metric Number Theory: the good and the bad

##### Abstract

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.

##### Authors

Thorn, Rebecca E##### Collections

- Theses [2542]