Suppose that $\mu$ is a probabilist measure on $\mathbb{Z}_p$ such that $d_{TV}(\mu,\mu_U)\leqslant \delta < 1$. What are the best upper bound on
$$ d_{TV}(\underbrace{\mu*\mu*\ldots*\mu}_{n \text{ times}},\mu)$$
Few remarks:
- One can show $2^{n-1}\delta^{n}$ which gives us the exponentially fast convergence for any $\delta<\frac{1}{2}$, for any abelian group.
- It always convergences to $0$. One can look at the problem as a random walk on $\mathbb{Z}_p$ with the transition matrix given by $Q(x,y)=\mu(y-x)$. From $\delta<1$ it follows that $\mu$ has a positive measure on some generator of $\mathbb{Z}_p$ and thus every state will be eventually visited.
- It seems (by some convexity considerations) that the worst case is when $\mu$ is $\delta$ at $0$ and uniform over some set of cardinality $(1-\delta)p$. Thus, problem reduces to the following question: let $B\subset\mathbb{Z}_p$ be of size $\delta p$ for $\delta>\frac{1}{2}$ (actually is the complement of the set above). Let $\mu_B$ be a uniform measure over $B$. What is the nontrivial bound on $$ \left| 2\mu_B - \mu_B*\mu_B \right|_{\ell_1(\mathbb{Z}_p)} $$ As one can easily see, a trivial bound is $4$.
- The claimed worst case corresponds to the following random walk: with probability $\delta$ the process doesn't change the state and otherwise shifts by a random element of $A$, where $|A|=\delta p$.
- (Additive combinatoric interpretation) Given the set $B\subset\mathbb{Z}_p$ of cardinality $\delta p$ what can we say about the minimal possible value of quantity $\sum\limits_{b\in B}r_{B,B}(b)$ where $r_{B,B}(x)$ counts the number of representations of $x$ as a sum of two elements of $B$.