Extensions to a Lemma of Bernik with Applications in the area of Metric Diophantine Approximation

This thesis is concerned with two extensions to a result of V. I Bernik [23] from 1983 which provides a quantitative description of the fact that two relatively prime polynomials in Z[x] cannot both have very small absolute values (in terms of their degrees and heights) in an interval unless that interval is extremely short. Bernik's result was presented for intervals in R and has the restriction that the polynomials being considered must have small modulus. In this thesis the result is extended to a cuboid in R3 and, in fact, it is clear from the proof that the result holds in Rn. Furthermore the restriction that the polynomials must have small modulus is removed. This is the first extension of Bernik's result to consider polynomials of large modulus. Bernik's result is also extended to a parallelepiped in R x C x Qp. This is not the first extension of this kind but the method of proof used leads to a new and very useful proposition