How Many Digits are Needed?

Let X1,X2,.. be the digits in the base-q expansion of a random variable X defined on [0, 1) where q≥2 is an integer. For n=1,2,.., we study the probability distribution Pn of the (scaled) remainder Tn(X)=∑k=n+1∞Xkqn-k: If X has an absolutely continuous CDF then Pn converges in the total variation metric to the Lebesgue measure μ on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that TN(X) follows μ and is independent of (X1,..,XN), and second exponentially fast convergence of Pn and its PDF fn. We discuss how many digits are needed and show examples of our results.