A 'word' is a finite or infinite sequence of symbols (called 'letters') taken from a finite non-empty set (called an 'alphabet'). In mathematics, words naturally arise when one wants to represent elements from some set (e.g., integers, real numbers, p-adic numbers, etc.) in a systematic way. For instance, expansions in integer bases (such as binary and decimal expansions) or continued fraction expansions allow us to associate with every real number a unique finite or infinite sequence of digits.

In this talk, I will discuss some old and new results in Combinatorics on Words and their applications to problems in Number Theory. In particular, by transforming inequalities between real numbers into (lexicographic) inequalities between infinite words representing their binary expansions, I will show how combinatorial properties of words can be used to completely describe the minimal intervals containing all fractional parts {x*2^n}, for some positive real number x, and for all non-negative integers n. This is joint work with Jean-Paul Allouche (Universite Paris-Sud, France).