Definitions and some motivation:
Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. Assume $\mu$ is non atomic and supp $\mu = X$.
Given a continuous function $f$ on $X$, we know by unique ergodicity that the Birkhoff averages $A_n f := \frac{1}{n}\sum_{k=0}^{n-1} T^k f$ converge uniformly to the constant function $Cf := \int_X f d\mu$. But how uniform is the convergence with respect to the convergence at other points?
Given a positive real valued continuous function $f$ on $X$, and $n \in \mathbb N$, define the error function $E_n: X \times \mathbb R^+ \to \mathbb R$ by
$E_n(x, r) := \frac{1}{\mu(B_r (x))} \int_{B_r (x)} |A_n f - Cf| d\mu$.
Define also for each $\delta > 0$, the set $S_\delta := \{ (x, r) | \ (x, r) \in X \times \mathbb R^+, \ \mu (B_r (x)) \geq \delta \}$.
Question: For fixed positive real valued continuous $f$, is it true that for all $\delta > 0$, we have $\ \limsup_{n \to \infty} \sup_{(x_1, r_1), (x_2, r_2) \in S_\delta} \frac{E_n (x_1, r_1) - E_n (x_2, r_2) }{E_n (x_1, r_1) + E_n (x_2, r_2)} = 0$?
Note: By convention we set $\frac{0}{0} = 0$.
