This book examines the number-theoretic properties of the real numbers. It collects a variety of new ideas and develops connections between different branches of mathematics. An indispensable compendium of basic results, the text also includes important theorems and open problems. The book begins with the classical results of Borel, Khintchine, and Weyl, and then proceeds to Diophantine approximation, GCD sums, Schmidt's method, and uniform distribution. Other topics include generalizations to higher dimensions and various non-periodic problems (for example, restricting approximation to fractions with prime numerator and denominator). It concludes with a chapter on Hausdorf dimensions for exceptional sets of measure zero.

Professor G. Harman, School of Mathematics, Mathematics Institute, University of Wales Cardiff, Senghennydd Road, Cardiff, CF2 4YH, email: Harman@cf.ac.uk

• Introduction


• 1: Normal numbers


• 2: Diophantine approximation


• 3: GCD sums with applications


• 4: Schmidt's method


• 5: Uniform distribution


• 6: Diophantine approximation with restricted numerator and denominator


• 7: Non-integer sequences


• 8: The integer parts of sequences


• 9: Diophantine approximation on manifolds


• 10: Hausdorff dimension of exceptional sets


• References


• Index