Exponents of Diophantine approximation and expansions in integer bases

Let ξ be a real number and let b ⩾ 2 be an integer. Let vb(ξ) or v′b(ξ) denote the supremum of the real numbers v for which the equation ‖bn ξ‖ ⩽ (bn)−v or ‖br (bs−1) ξ‖ ⩽ (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 (ξ) denote the supremum of the real numbers v for which the equation ‖q ξ‖ < 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.