Mixing time of the Chung--Diaconis--Graham random process

Define $(X_n)$ on $\mathbf{Z}/q\mathbf{Z}$ by $X_{n+1} = 2X_n + b_n$, where the steps $b_n$ are chosen independently at random from $-1, 0, +1$. The mixing time of this random walk is known to be at most $1.02 \log_2 q$ for almost all odd $q$ (Chung--Diaconis--Graham, 1987), and at least $1.004 \log_2 q$ (Hildebrand, 2008). We identify a constant $c = 1.01136\dots$ such that the mixing time is $(c+o(1))\log_2 q$ for almost all odd $q$. In general, the mixing time of the Markov chain $X_{n+1} = a X_n + b_n$ modulo $q$, where $a$ is a fixed positive integer and the steps $b_n$ are i.i.d. with some given distribution in $\mathbf{Z}$, is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a $1+o(1)$ factor whenever the entropy exceeds $(\log a)/2$.

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