No abstract

We study a random walk on Fp defined by Xn+1 = 1/Xn + εn+1 if Xn = 0, and Xn+1 = εn+1 if Xn = 0, where εn+1 are independent and identically distributed. This can be seen as a non-linear analogue of the Chung-Diaconis-Graham process. We show that the mixing time is of order log p, answering a question of Chatterjee and Diaconis [12].

No abstract

The list Ramsey number R ℓ (H, k), recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if H is a fixed r-uniform hypergraph that is not r-partite and the number of colors k goes to infinity, e Ω( k) . We prove that R ℓ (H, k) = e Θ(k) if and only if H is not r-partite.

We show that if f is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every x is at least $1-10^{-45}$, while also strictly smaller than 1. For large x, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where C is some positive constant.