  A subset $A$ of $\nats$ is called an IP-set if $A$ contains all finite sums
of distinct terms of some infinite sequence $(x_n)_{n\in \nats} $ of natural
numbers. Central sets, first introduced by Furstenberg using notions from
topological dynamics, constitute a special class of IP-sets possessing rich
combinatorial properties: Each central set contains arbitrarily long arithmetic
progressions, and solutions to all partition regular systems of homogeneous
linear equations. In this paper we investigate central sets in the framework of
combinatorics on words. Using various families of uniformly recurrent words,
including Sturmian words, the Thue-Morse word and fixed points of weak mixing
substitutions, we generate an assortment of central sets which reflect the rich
combinatorial structure of the underlying words. The results in this paper rely
on interactions between different areas of mathematics, some of which had not
previously been directly linked. They include the general theory of
combinatorics on words, abstract numeration systems, and the beautiful theory,
developed by Hindman, Strauss and others, linking IP-sets and central sets to
the algebraic/topological properties of the Stone-\v{C}ech compactification of
$\nats .$
