IDEAS AND RESULTS FROM THE THEORY OF DIOPHANTINE APPROXIMATION

In many areas ofmathematics problems of small divisors, or exceptional sets on which certain desired qualities do not hold, appear. The obvious question that then arises is how large are these exceptional sets? This question leads to other questions regarding what do we mean by size. For example there exist many sets of Lebesgue measure zero which have positive Hausdorff dimension implying that although small they are still uncountable. Similarly how does one compare two sets of the same Hausdorff dimension – recent results using Hausdorff measure are one possibility. Diophantine approximation began as a study of how closely real num-bers could be approximated by rationals. The aim of this paper is to show how the classical results of real Diophantine approximation have been adapted and extended to deal with other kinds of approximation in other spaces. Three cases will be specifically discussed, those being the classical $\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}^{-},\cdot$. approximation by algebraic numbers and Diophan