DIOPHANTINE ANALYSIS AND WORDS
Iekata Shiokawa, Michel Waldschmidt, Christian Christian, Shiokawa, Iekata, Tamura, Jun-ichi Rauzy’s · 2006
There is no explicitly known example of a triple (g, a, x), where g ≥ 3 is an integer, a a digit in {0, . . . , g − 1} and x a real algebraic irrational number, for which one can claim that the digit a occurs infinitely often in the g–ary expansion of x. In 1909 and later in 1950, E. Borel considered such questions and suggested that the g–ary expansion of any algebraic irrational number in any base g ≥ 2 satisfies some of the laws that are satisfied by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress have been made recently, mainly thanks to clever use of the Schmidt’s subspace Theorem. We review some of these results.