A Multiplicatively Symmetrized Version of the Chung-Diaconis-Graham Random Process

This paper considers random processes of the form $$X_{n+1}=a_nX_n+b_n\pmod p$$ where p is odd, $$X_0=0$$ , $$(a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots $$ are i.i.d., and $$a_n$$ $$b_n$$ independent with $$P(a_n=2)=P(a_n=(p+1)/2)=1/2$$ $$P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$$ . can be viewed as a multiplicatively symmetrized version process Chung, Diaconis, Graham. shows that order $$(\log p)^2$$ steps suffice for $$X_n$$ to close uniformly distributed on integers mod all odd while necessary p.