A Generalised abc Conjecture and Quantitative Diophantine Approximation

The abc Conjecture and its number field variant have huge implications across a wide
range of mathematics. While the conjecture is still unproven, there are a number of
partial results, both for the integer and the number field setting. Notably, Stewart
and Yu have exponential abc bounds for integers, using tools from linear forms in
logarithms, while Győry has exponential abc bounds in the number field
case, using methods from S-unit equations [20]. In this thesis, we aim to combine
these methods to give improved results in the number field case. These results are
then applied to the effective Skolem-Mahler-Lech problem, and to the smooth abc
conjecture.

The smooth abc conjecture concerns counting the number of solutions to a+b = c
with restrictions on the values of a, b and c. this leads us to more general methods
of counting solutions to Diophantine problems. Many of these results are asymptotic
in nature due to use of tools such as Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory". We make these
lemmas effective rather than asymptotic other than on a set of size δ > 0, where δ is
arbitrary. From there, we apply these tools to give an effective Schmidt’s Theorem,
a quantitative Koukoulopoulos-Maynard Theorem (also referred to as the Duffin-
Schaeffer Theorem), and to give effective results on inhomogeneous Diophantine
Approximation on M0-sets, normal numbers and give an effective Strong Law of
Large Numbers. We conclude this thesis by giving general versions of Lemmas 1.4
and 1.5 of Harman's "Metric Number Theory".