On the multiplicative Chung-Diaconis-Graham process

Abstract

We study the lazy Markov chain on $\mathbb{F}_p$ defined as follows: $X_{n+1}=X_n$ with probability $1/2$ and $X_{n+1}=f(X_n) \cdot \varepsilon_{n+1}$, where the $\varepsilon_n$ are random variables distributed uniformly on the set $\{\gamma, \gamma^{-1}\}$, $\gamma$ is a primitive root and $f(x)=x/(x-1)$ or $f(x)=\mathrm{ind}(x)$. Then we show that the mixing time of $X_n$ is $\exp(O(\log p \cdot \log \log \log p/ \log \log p))$. Also, we obtain an application to an additive-combinatorial question concerning a certain Sidon-type family of sets. Bibliography: 34 titles.