On the definition of quasi-generic point In Furstenberg's book "Recurrence in Ergodic Theory and Combinatorial Number Theory" the notion of quasi-generic point is defined as follows:
Let $X$ be a compact metric space, $T$ a continuous transformation of $X$, and $\mu$ an invariant measure. We say $x_0\in X$ is quasi-generic for $\mu$ if for some sequence of pairs of integers $\{a_k,b_k\}$ with $a_k\leq b_k$ and $b_k-a_k\to\infty$ we have $$\frac{1}{b_k-a_k+1}\sum_{n=a_k}^{b_k} f(T^nx_0)\rightarrow \int f\ d\mu$$ as $k\to \infty$, for every continuous function $f\in C(X)$.
While in several other books, for example, Glasner's book "Ergodic Theory via Joining" the definition of quasi-generic point is as the above definition but 
$a_k$ is fixed, always equal to $0$.
So my question is that are the two definitions the same? As they have the same name I guess the two definitions are the same, but I am sure how to compare these averages, since $a_k$ may grow much faster than $b_k-a_k$.
 A: I don't think the two definitions are the same.
Take, for instance, the one-sided binary full shift $X:=\{0,1\}^{\mathbb{N}}$ with $(Tx)(i):=x(i+1)$.  Let $a_0:=0$ and recursively define $b_k:=a_k+k$ and $a_{k+1}:=b_k+2^k$.
Define $x_0\in X$ by
\begin{align*}
   x_0(i) &:=
      \begin{cases}
         1 & \text{if $a_k\leq i<b_k$,} \\
         0 & \text{if $b_k\leq i<a_{k+1}$.}
      \end{cases}
\end{align*}
Then, $x_0$ is quasi-generic for $\delta_{\underline{1}}$ (i.e., point mass at the fixed point $\underline{1}$) in the first sense but not in the second sense.
Namely, let $f:X\to\mathbb{R}$ be a local function where $f(x)$ depends only on symbols $x_0,x_1,\ldots,x_{r-1}$.  (Recall: local functions are dense in $C(X)$, so we only need to verify the ergodic equality for local functions.)
Then, 
\begin{align*}
   \sum_{a_k\leq i<b_k} f(T^ix_0)
   &= (k-r)f(\underline{1}) + \sum_{b_k-r\leq i<b_k} f(T^ix_0) \\
   &= k f(\underline{1}) + o(k) \qquad\text{as $k\to\infty$,}
\end{align*}
which implies that $x_0$ is quasi-generic for $\delta_{\underline{1}}$.
On the other hand, the point $x_0$ is generic for $\delta_{\underline{0}}$, because
\begin{align*}
   \sum_{0\leq i<m} f(T^ix_0)
   &= m f(\underline{0}) + o(m) \qquad\text{as $m\to\infty$.}
\end{align*}
(Note that $\bigcup_k\{a_k,a_k+1,\ldots,b_k-1\}$ has density $0$ in $\mathbb{N}$.)
According to the second definition, if a point is generic for a measure $\mu$, it cannot be quasi-generic for another measure.
