= Solution
Use the probability-system convention $\mu(X)=1$. The <Birkhoff ergodic theorem>, also called the <pointwise ergodic theorem>, states that for a <measure-preserving system> and $f\in L^1(\mu)$,
$$
A_Nf(x)=\frac1N\sum_{n=0}^{N-1}f(T^nx)
\longrightarrow\mathbb E_\mu[f\mid\mathcal I](x)
\quad\text{almost everywhere},
$$
where $\mathcal I$ is the <invariant sigma-algebra>. The limit is integrable and has the same integral as $f$. On a probability space the convergence also holds in $L^1$, as in the allowed mean ergodic theorem. If $T$ is an <ergodic transformation>, $\mathcal I$ is trivial modulo null sets, giving
$$
\boxed{A_Nf(x)\longrightarrow\int_X f\,d\mu\quad\text{almost everywhere}.}
$$
The <integer multiplication map on the circle> preserves <Lebesgue measure>: for any integrable $g$ on $[0,1)$,
$$
\int_0^1g(Kx\bmod1)\,dx
=\sum_{j=0}^{K-1}\int_{j/K}^{(j+1)/K}g(Kx-j)\,dx
=\int_0^1g(y)\,dy.
$$
To prove the <ergodicity of integer multiplication on the circle>, suppose $u\in L^2(m)$ satisfies $u\circ T_K=u$. Let $c_j$ be its <Fourier coefficients> in the <Fourier basis> $e_j(x)=e^{2\pi ijx}$. Since $e_j\circ T_K=e_{Kj}$, the <Fourier coefficients> of $u\circ T_K$ at index $j$ are zero if $K$ does not divide $j$, and are $c_{j/K}$ otherwise. This identity holds for all $L^2$ functions by approximation with <trigonometric polynomials> and the isometry $u\mapsto u\circ T_K$. Invariance gives
$$
c_j=\begin{cases}c_{j/K},&K\mid j,\\0,&K\nmid j.\end{cases}
$$
Every nonzero integer $j$ can be divided by $K$ only finitely often. Thus $c_j=0$ for every $j\ne0$, and completeness of the <Fourier basis> makes $u$ constant <almost everywhere>. Applying this to the <indicator function> of an invariant set gives measure zero or one, so
$$
\boxed{T_K\text{ is ergodic for }m\quad(K\geq2).}
$$
A <normal number> in base $K$ has every word $w=(w_1,\ldots,w_r)$ of $r$ base-$K$ digits occurring with limiting overlapping frequency $K^{-r}$. Use the expansion that is not eventually equal to $K-1$ when there are two expansions. The word $w$ corresponds to the half-open interval
$$
I_w=\left[\frac{j(w)}{K^r},\frac{j(w)+1}{K^r}\right),\qquad
j(w)=\sum_{i=1}^r w_iK^{r-i}.
$$
A word starting at position $n+1$ occurs exactly when $T_K^nx\in I_w$. The <Birkhoff ergodic theorem>, applied to $\mathbf1_{I_w}$, gives frequency $m(I_w)=K^{-r}$ <almost everywhere>. There are countably many pairs $(K,w)$, so their full-measure sets have a full-measure intersection. In particular,
$$
\boxed{m\{x\in[0,1):x\text{ is normal in every integer base }K\geq2\}=1.}
$$
These are <absolutely normal numbers>, so existence follows as well. This interval description also proves <normality and equidistribution under integer multiplication>: the base-$K$ intervals form arbitrarily fine grids, so their frequencies imply the correct frequency for every interval by approximation from inside and outside.
For the growth assertion, put $g=|f|$. For every $\varepsilon>0$, the <Tonelli theorem> gives the useful summability bound
$$
\sum_{n=1}^\infty\mu(g>\varepsilon n)
=\int_X\sum_{n=1}^\infty\mathbf1_{\{g(x)>\varepsilon n\}}\,d\mu(x)
\leq\frac{\|f\|_1}{\varepsilon}.
$$
Since $T$ is a <measure-preserving transformation>, $\mu(g\circ T^n>\varepsilon n)=\mu(g>\varepsilon n)$. The first <Borel-Cantelli lemma> shows that $g(T^nx)>\varepsilon n$ occurs only finitely often <almost everywhere>. Intersecting the resulting full-measure sets for $\varepsilon=1,1/2,1/3,\ldots$ proves the <linear growth bound for integrable observables>, $f(T^nx)/n\to0$. Multiplication by $n^{1-a}\leq1$ then gives
$$
\boxed{\frac{f(T^nx)}{n^a}\longrightarrow0\quad\text{almost everywhere, for every }a\geq1.}
$$
The threshold is sharp. For $0<a<1$, choose $p$ with $1<p<1/a$, and take the <Bernoulli shift> on $X=(0,1)^{\mathbb Z}$ with the <product measure> of independent uniform coordinates. The left shift preserves that measure because it preserves the probability of every finite-coordinate event. Define $f(x)=x_0^{-1/p}$; it is integrable because
$$
\int_X f\,d\mu=\int_0^1u^{-1/p}\,du=\frac{p}{p-1}<\infty.
$$
The variables $f(T^nx)=x_n^{-1/p}$ are independent. For any fixed $M\geq1$,
$$
\mu\{f(T^nx)>Mn^a\}=M^{-p}n^{-ap},\qquad ap<1.
$$
The probability sum diverges, so the second <Borel-Cantelli lemma> makes these events occur infinitely often <almost surely>. Intersecting over positive integer $M$ even yields $\limsup_n f(T^nx)/n^a=\infty$. For $a\leq0$, the constant observable $f=1$ already fails to give limit zero. Thus the <sharpness of the linear growth bound for integrable observables> gives
$$
\boxed{\text{There is no }a<1\text{ for which the asserted limit holds in every system for every }f\in L^1.}
$$
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