Number theory, one of the oldest branches of mathematics, and dynamical systems, one of the newest, are rather disparate and might well be expected to have little in common. There are however many surprising connections between them. One emerged last century from the study of the stability of the solar system where the problem of “small divisors or denominators” associated with the near resonance of planetary frequencies arises and which made convergence in the series solution highly problematic. The phenomenon of “small divisors” is closely related to Diophantine approximation and it is perhaps no coincidence that Dirichlet, Kronecker and Siegel all worked on small divisor problems. These proved quite intractable until relatively recently (1942), when Siegel used ideas drawn from the theory of Diophantine approximation to overcome a problem of small divisors arising in the iteration of analytic functions near a fixed point [1]. Twenty years ago the question of the stability of the solar system was answered in more general terms by the celebrated Kolmogorov-Arnol'd-Moser theorem ([2], Appendix 8). The corresponding small divisor problem is dealt with by using SiegePs idea of imposing a suitable Diophantine inequality on the frequencies to ensure that they are not too close to resonance. Thus here Diophantine approximation again plays a central role [3].

The connection between resonance and Diophantine equations and near-resonance and Diophantine approximation is (with hindsight) a natural one. But there are other connections in quite different settings. Szemeredi's theorem that any infinite integer sequence of positive upper density contains arbitrarily long arithmetic progressions has been proved by Furstenberg using ideas from dynamical systems [4].