On the multiplicative Chung–Diaconis–Graham process

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Abstract

We study the lazy Markov chain on Fp">Fp defined as follows: Xn+1=Xn">Xn+1=Xn with probability 1/2">1/2 and Xn+1=f(Xn)⋅εn+1">Xn+1=f(Xn)⋅εn+1, where the εn">εn are random variables distributed uniformly on the set {γ,γ−1}">{γ,γ−1}, γ">γ is a primitive root and f(x)=x/(x−1)">f(x)=x/(x−1) or f(x)=ind(x)">f(x)=ind(x). Then we show that the mixing time of Xn">Xn is exp⁡(O(log⁡p⋅log⁡log⁡log⁡p/log⁡log⁡p))">exp⁡(O(log⁡p⋅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.

About the authors

Ilya Dmitrievich Shkredov

Steklov Mathematical Institute of Russian Academy of Sciences

Author for correspondence.
Email: ilya.shkredov@gmail.com
Doctor of physico-mathematical sciences, Professor

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