Resumen de Exponents of Diophantine approximation and expansions in integer bases

Masaaki Amou, Yann Bugeaud

• Let î be a real number and let b  2 be an integer. Let vb(î) or v  b(î) denote the supremum of the real numbers v for which the equation bnî  (bn) .v or br(bs . 1)î  (br+s) .v has infinitely many solutions in positive integers n or r and s, respectively. Here,  ·  stands for the distance to the nearest integer. Also let v1(î) denote the supremum of the real numbers v for which the equation qî < q .v has infinitely many solutions in positive integers q. Motivated by the question whether one can read the irrationality exponent of a real number on its b-ary expansion, we establish various results on the set of values taken by the triple of functions (v1, vb, v  b) evaluated at real points.