The sequence
{nα}, where α is an irrational number and {.} denotes fractional part, plays
a fundamental role in probability theory,
analysis and number theory. For suitable α, this sequence provides an example
for "most uniform" infinite sequences, i.e. sequences whose
discrepancy has the
smallest possible order of magnitude. Such 'low
discrepancy' sequences have important applications in Monte Carlo integration
and other problems of numerical mathematics. For rapidly increasing nk
the behaviour of {nkα} is similar to that of independent random
variables, but its asymptotic properties depend strongly also on the number
theoretic properties of nk, providing a simple example for
pseudorandom behaviour. Finally, for periodic f the sequence f(nα) provides a
generalization of the trig-onometric system with many interesting properties. In this lecture, we give a survey
of the field (going back more than 100
years) and formulate new results.