Is it possible to distinguish algebraic from transcendental real numbers by
considering the b-ary expansion in some base b≥2? In 1950, \'E. Borel
suggested that the answer is no and that for any real irrational algebraic
number x and for any base g≥2, the g-ary expansion of x should
satisfy some of the laws that are shared by almost all numbers. 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 irrational algebraic number, for
which one can claim that the digit a occurs infinitely often in the g-ary
expansion of x. However, some progress has been made recently, thanks mainly
to clever use of Schmidt's subspace theorem. We review some of these results