= Solution
Use the convention that level $k$ counts the $k$ nonzero shifts, hence an <arithmetic progression> of length $k+1$. The <SZ property> at level $k$ means that every measurable set $A$ of positive measure satisfies
$$
\boxed{\liminf_{N\to\infty}\frac1N\sum_{n=1}^N\mu\bigl(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-kn}A\bigr)>0.}
$$
This is the averaged multiple-recurrence property, stronger than merely finding one positive intersection. Shifting the indexing convention from $k+1$ terms to $k$ terms does not change the all-levels assertion. We work with probability <measure-preserving systems>.
Here is the <Furstenberg correspondence principle> construction proving the <Szemerédi theorem>. Let $D\subseteq\mathbb N$ have positive upper density $\delta=\limsup_N|D\cap\{1,\ldots,N\}|/N$. Put $z\in\{0,1\}^{\mathbb Z}$ equal to the indicator of $D$ on the positive coordinates and zero elsewhere, and let $S$ be the <left shift>. The binary <full shift> $\Omega=\{0,1\}^{\mathbb Z}$ is a compact <metric space> in the <product topology>, and $S$ is a <homeomorphism>. Choose $N_i\to\infty$ attaining the upper density and define the <empirical measures>
$$
\mu_i=\frac1{N_i}\sum_{m=1}^{N_i}\delta_{S^mz}.
$$
We use the standard weak <compactness> fact that every sequence of <Borel probability measures> on a compact <metric space> has a weakly convergent subsequence, where weak convergence means convergence of integrals of continuous functions. Pass to a subsequence with limit $\mu$. For every continuous $F$, telescoping gives
$$
\int F\circ S\,d\mu_i-\int F\,d\mu_i
=\frac{F(S^{N_i+1}z)-F(Sz)}{N_i}\longrightarrow0.
$$
Passing to the limit proves $S_*\mu=\mu$. The <cylinder set> $A=\{w:w(0)=1\}$ is clopen, so its indicator is continuous and
$$
\mu(A)=\lim_i\mu_i(A)=\delta>0.
$$
If every <measure-preserving system> has the <SZ property> at level $k$, there is some $n\geq1$ for which $\mu(C_n)>0$, where $C_n=\bigcap_{j=0}^kS^{-jn}A$. This is also a clopen <cylinder set>, so $\mu_i(C_n)\to\mu(C_n)>0$. For some sufficiently large $i$, at least one term of the empirical average is positive: $S^mz\in C_n$ for some $1\leq m\leq N_i$. Unwinding the coordinates gives
$$
z(m)=z(m+n)=\cdots=z(m+kn)=1.
$$
Therefore \b[$D$ contains an <arithmetic progression> of length $k+1$ with positive common difference]. Having the <SZ property> at every level gives the infinite-density form of the <Szemerédi theorem>.
The finite uniform form follows from the same argument. If, for fixed $\delta>0$ and $k$, arbitrarily large $N_i$ admitted sets $D_i\subseteq\{1,\ldots,N_i\}$ with $|D_i|\geq\delta N_i$ and no such <arithmetic progression>, replace $z$ above by the indicator $z_i$ of $D_i$, extended by zero outside that interval. Their empirical measures still have an invariant weak limit with $\mu(A)\geq\delta$. For every fixed positive $n$, however, $\mu_i(C_n)=0$, since a point contributing to that cylinder would be an actual progression in $D_i$. Thus $\mu(C_n)=0$ for all positive $n$, contradicting the <SZ property>. Consequently a sufficiently large $N$ depending only on $\delta,k$ forces the required progression.
A <compact measure-preserving system> is one for which every observable $f\in L^2(\mu)$ has relatively compact forward orbit $\{U^nf:n\geq0\}$ under the <Koopman operator> $Uf=f\circ T$. Equivalently, each of these orbits is totally bounded in the global $L^2$ norm; <compactness> refers to observables, not just to the underlying topological space. For an invertible system it is usual to use all integer times. One may also define <compactness> by density of <almost periodic observables>: the two definitions coincide because $\|U^n(f-g)\|_2=\|f-g\|_2$, so a finite net for an approximating observable is a finite net, with slightly larger radius, for $f$.
To prove the <SZ property>, fix $A$ with $a=\mu(A)>0$, put $f=\mathbf1_A$, and choose
$$
\varepsilon^2<\frac{a}{2\sum_{j=1}^kj^2}.
$$
<Compactness> gives a finite $\varepsilon$-net for its orbit with centers $U^{r_1}f,\ldots,U^{r_s}f$, where $r_i\geq0$. Let $M=\max_i r_i$. For every integer $n\geq M+1$, some center satisfies $\|U^nf-U^{r_i}f\|_2<\varepsilon$. Since $U$ is an <isometry>, cancellation gives
$$
\|U^{n-r_i}f-f\|_2<\varepsilon.
$$
Thus the return set $R=\{r\geq1:\|U^rf-f\|_2<\varepsilon\}$ meets every interval $\{n-M,\ldots,n\}$ for $n\geq M+1$. In particular, $R$ is a <syndetic set> with lower density at least $1/(M+1)$. This proves the needed <syndetic near returns of an almost periodic observable> without any assumption of <ergodicity>.
For $r\in R$, telescoping and the <isometry> property give
$$
\|U^{jr}f-f\|_2\leq\sum_{\ell=0}^{j-1}\|U^{\ell r}(U^rf-f)\|_2<j\varepsilon\qquad(1\leq j\leq k).
$$
As $f$ is an indicator, $\|U^{jr}f-f\|_2^2=\mu(A\mathbin\triangle T^{-jr}A)$. The <union bound> therefore gives
$$
\begin{aligned}
\mu\left(\bigcap_{j=0}^kT^{-jr}A\right)
&\geq a-\sum_{j=1}^k\mu(A\setminus T^{-jr}A)\\
&\geq a-\sum_{j=1}^k\|U^{jr}f-f\|_2^2
>a-\varepsilon^2\sum_{j=1}^kj^2>\frac a2.
\end{aligned}
$$
All other multiple-intersection measures are nonnegative. Averaging over the bounded-gap set $R$ yields
$$
\boxed{\liminf_{N\to\infty}\frac1N\sum_{r=1}^N\mu\left(\bigcap_{j=0}^kT^{-jr}A\right)\geq\frac{a}{2(M+1)}>0.}
$$
Thus \b[every compact <measure-preserving system> has the <SZ property> at every level], including nonergodic systems and, under the forward-orbit definition, noninvertible transformations.
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