Exponents of Diophantine approximation and expansions in integer bases

Let ξ be a real number and let b ⩾ 2 be an integer. Let v b ( ξ ) or v ′ b ( ξ ) denote the supremum of the real numbers v for which the equation ‖ b n ξ ‖ ⩽ ( b n ) − v or ‖ b r ( b s −1) ξ ‖ ⩽ ( b r + 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 v 1 ( 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 ( v 1 , v b , v ′ b ) evaluated at real points.