In 1870 Georg Cantor proved that a 2sr periodic omplex valued function of a real variable coincides with the values of at most one trigonometric series. We present his proof and then survey some of the many one dimensional generalizations and extensions of Cantor's theorem. We also survey the situation in higher dimensions, where a great deal less is known. 1. Cantor's uniqueness theorem. In 18...

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. 2000AMS subject classification: 37K55, 35L05.

In a recent paper, Baker and Kong have studied the Hausdorff dimension of intersection Cantor sets with their translations. We extend results to more general sets. The proofs rely on frequencies digits in unique expansions non-integer bases. relation this, we introduce practical method determine frequency any given finite block Thue--Morse type sequences.