One hundred years of normal numbers
by
Glyn Harman
Royal Holloway, University of London

The study of the arithmetic properties of almost all numbers (in the sense of Lebesgue measure on R) began in the early years of the twentieth century with the researches of Borel. The general problem is to use a real number to define an infinite sequence of integers (via its decimal expansion or continued fraction expansion, for example) and to analyse the arithmetic properties of the sequence. In this talk we begin with some classical work of Borel, follow some of its development through the twentieth century, and finish with some new results where several sequences are generated and their properties analysed simultaneously (for example: n, [n\alpha], [(n \alpha)²], ... , [(n \alpha)^k] simultaneously prime infinitely often for almost all \alpha > 0).

Date received: February 28, 2000