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For a Feller semigroup on a compact metric space, time-averaged transition laws have weakly convergent subsequences by compactness of probability measures on a compact metric space. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.

On a compact metric space $X$, a conservative Feller semigroup is a strongly continuous semigroup of positive linear maps on $C(X)$ satisfying $P_{t}1=1$. Positivity means $f\ge 0⟹P_{t}f\ge 0$. These conditions imply sup-norm contraction. Conversely, for a contraction semigroup with $1\in D(L)$ and $L1=0$, differentiation on the generator domain gives $\frac{d}{dt}{P}_{t}1=P_{t}L1=0$, so $P_{t}1=1$. On the real space $C(X)$, if $0\le f\le 1$, then $\Vert 1-P_{t}f{\Vert }_{\infty }\le \Vert 1-f{\Vert }_{\infty }\le 1$, implying $P_{t}f\ge 0$. Scaling proves positivity for every nonnegative $f$. This is the unital contraction positivity criterion. For complex $C(X)$ the same conclusion follows because each functional $f\mapsto P_{t}f(x)$ has norm one and value one on $1$, hence is a positive linear functional.

For each $t,x$, the Riesz-Markov-Kakutani representation theorem represents that functional by a unique Borel probability measure $p_t(x,dy)$:Continuity of $P_{t}f$ gives weak continuity of $x\mapsto p_t(x,\cdot )$. Approximating indicators of open sets increasingly by continuous functions proves Borel measurability in $x$; a monotone-class argument then gives a Markov kernel. The semigroup property and uniqueness of representing measures yield $p_0(x,\cdot )={\delta }_x$ and the Chapman-Kolmogorov equationThese are the required transition probabilities.

An invariant probability measure for a semigroup satisfies ${\int }_X{P}_{t}fd\mu ={\int }_{X}fd\mu$ for every $t\ge 0$ and $f\in C(X)$. Equivalently its distribution is preserved by the Markov kernel. For $1\le p<\infty$, Jensen inequality gives $|P_{t}f{|}^p\le P_t(|f{|}^p)$ pointwise. Integrating and using invariance provesAlso $\Vert P_{t}f-f{\Vert }_{L^p(\mu )}\le \Vert P_{t}f-f{\Vert }_{\infty }\to 0$. Continuous functions are dense in these $L^p$ spaces for a finite Borel measure on a compact metric space, so the contraction extends to a strongly continuous semigroup on $L^p(\mu )$.

For real $f$ with $f,f^2\in D(L)$, the kernel form of Jensen inequality gives $P_t(f^2)-(P_{t}f{)}^2\ge 0$, with equality at $t=0$. Take its right derivative to obtain the generator square inequalityThis is nonnegativity of the associated carré du champ.

Each time-average functional ${\varphi }_n$ is positive with ${\varphi }_n(1)=1$, hence represents a Borel probability measure ${\mu }_n$. The compactness of probability measures on a compact metric space gives a subsequence converging weakly to a probability measure $\mu$. Explicitly, $C(X)$ is a separable Banach space, so Banach-Alaoglu theorem and a countable dense family give a subsequence converging on every continuous function, rather than merely a net. For fixed $t\ge 0$, the semigroup property gives the Krylov-Bogolyubov time-average argument:Its absolute value is at most $2t\Vert f{\Vert }_{\infty }/n$. Pass to the subsequential limit, observing that $P_{t}f\in C(X)$, to get ${\int }_X{P}_{t}fd\mu ={\int }_{X}fd\mu$. Thus the limiting measure is invariant.

The Szemerédi theorem says that for every integer $k\ge 2$ and every $\delta >0$, there is $N_0(k,\delta )$ such that, for $N\ge N_0$, every $A\subseteq \{1,\dots ,N\}$ with $|A|\ge \delta N$ contains a nonconstant arithmetic progressionEquivalently, every subset of the positive integers with positive upper asymptotic density contains arithmetic progressions of every finite length. Length one is immediate.

The Furstenberg multiple recurrence theorem states that for every probability measure-preserving system, every measurable $B$ with $\mu (B)>0$, and every integer $k\ge 2$, there is $n\ge 1$ withHere $T^{-jn}B$ means a preimage, so invertibility is unnecessary. The Furstenberg multiple recurrence theorem also does not assume an ergodic transformation.

We prove the implication to the finite Szemerédi theorem by constructing the Furstenberg correspondence principle explicitly. Suppose, to the contrary, that for fixed $k$ and $\delta >0$ there are $N_{\ell }\to \infty$ and $A_{\ell }\subseteq \{1,\dots ,N_{\ell }\}$ with $|A_{\ell }|\ge \delta N_{\ell }$, but with no length-$k$ arithmetic progression of positive common difference. In the binary full shift $\Omega =\{0,1{\}}^{\mathbb{Z}}$, encode $A_{\ell }$ as $x_j^{(\ell )}=1_{A_{\ell }}(j)$, taking all coordinates outside $A_{\ell }$ to be zero. Let $S$ be the left shift, $(S\omega {)}_j={\omega }_{j+1}$, and setThe cylinder set $C$ is a clopen set, and ${\nu }_{\ell }(C)=|A_{\ell }|/N_{\ell }\ge \delta$. The binary full shift is a compact metric space, so compactness of probability measures on a compact metric space gives a subsequence of these empirical measures with weak convergence of probability measures to a Borel probability measure $\nu$.

For every continuous $h$ on the full shift,Passing to the weak limit shows that $\nu$ is an invariant measure for the continuous left shift. Thus $(\Omega ,\mathcal{B},\nu ,S)$ is a probability measure-preserving system. Since the indicator function of $C$ is continuous, $\nu (C)\ge \delta$.

Apply the Furstenberg multiple recurrence theorem to $C$. For some $n\ge 1$, the clopen sethas $\nu (D)>0$. Its continuous indicator function gives ${\nu }_{\ell }(D)\to \nu (D)>0$. For large $\ell$, at least one $S^a{x}^{(\ell )}$ in the defining empirical measure therefore belongs to $D$. The choice of left shift impliesso $a,a+n,\dots ,a+(k-1)n$ lies in $A_{\ell }$. This contradicts the assumed absence of arithmetic progressions and proves the finite Szemerédi theorem. Applying the finite Szemerédi theorem on intervals where an infinite set has density bounded below proves the stated upper asymptotic density formulation.

For multiple recurrence for circle rotations, write $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $R_{\alpha }x=x+\alpha$, with normalized Lebesgue measure $m$, which is the Haar measure of the circle group. Fix a measurable $B\subseteq \mathbb{T}$ with $m(B)>0$ and $k\ge 2$. If $\alpha$ is rational, there is $q\ge 1$ with $R_{\alpha }^q$ the identity, and $n=q$ gives an intersection of measure $m(B)$.

For irrational $\alpha$, the pigeonhole principle applied to $0,\alpha ,\dots ,Q\alpha$ in $Q$ equal arcs gives $1\le n\le Q$ with $\Vert n\alpha {\Vert }_{\mathbb{T}}\le 1/Q$. In particular there are arbitrarily small nonzero returns to zero for the irrational rotation of the circle.

We also need translation continuity in L1 on the circle. Given a measurable $B$, approximate its indicator function in the L1 norm by a continuous $g$ on the circle group. Such approximation follows from regularity of Lebesgue measure, or approximation by finite unions of intervals. Translation invariance and uniform continuity givewhere the approximation error is first made arbitrarily small. Equivalently, $m(B△(B-t))\to 0$.

Choose a return time $n\ge 1$ so small modulo one that for each $1\le j<k$,This is possible because multiplication by each fixed $j$ is continuous on the circle group and translation continuity in L1 on the circle applies to the finitely many translates. The union bound now givesThis proves the Furstenberg multiple recurrence theorem for every circle rotation with its normalized Lebesgue measure, including both rational and irrational angles.