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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.

We give a spectral proof of square-difference recurrence for finite colourings, proving the auxiliary results it uses. Extend the colouring arbitrarily to nonpositive integers, and let $f_j$ be the indicator of colour $j$. By successively taking subsequences of the interval lengths, followed by a diagonal subsequence, we may arrange that all the limitsexist, for every integer $h$. This uses only sequential compactness of bounded real sequences: list the countably many quantities, choose a convergent subsequence for each in turn, and take the diagonal. All limits below are along this common subsequence. We have $c(0)=1$, $c(-h)=c(h)$, $c(h)\ge 0$ and ${\sum }_j{\delta }_j=1$. Translating a fixed interval endpoint changes a normalized average by $O(|h|/L)$.

The sequence is a positive-definite function on $\mathbb{Z}$. Indeed, for finitely many complex numbers $z_u$,We next prove the required Bochner–Herglotz theorem for this sequence. The trigonometric polynomialis nonnegative by positive definiteness, and its integral on the circle is $c(0)=1$. Thus $F_H(t)dt$ is a probability measure. Such measures on a compact interval have a weakly convergent subsequence. For completeness, take a diagonal subsequence on all rational points of their nondecreasing distribution functions. The right-continuous extension of the limiting rational values is a distribution function and defines a probability measure by its interval increments. At its continuity points the original distribution functions converge, by sandwiching between rational endpoints. Approximating continuous functions by step functions with such endpoints proves convergence of their integrals. Identify the interval endpoints to obtain a measure on the circle. This proves the compactness of probability measures on a compact metric space in the case needed here.

For a fixed $h$, the Fourier coefficient of $F_{H}dt$ is $(1-|h|/H)c(h)$ once $H>|h|$. Taking the weak limit therefore supplies a positive probability measure $\mu$ such thatNo spectral representation has been assumed without proof.

Its atom at zero has positive mass. For each $H$,The last inequality is Cauchy-Schwarz on each colour's interval average. The integrand is at most one and tends to zero for every nonzero point of the circle, by the geometric-series formula, while it is one at zero. Bounded convergence gives

We also prove that quadratic phases with an irrational coefficient average to zero. For $z_0,\dots ,z_{L-1}$ bounded by one, extended by zero, Cauchy-Schwarz applied to ${\sum }_x{\sum }_{u=0}^{H-1}{z}_{x+u}=H{\sum }_x{z}_x$ yieldsThis identity follows by expanding the square of the inner shifted sum and grouping pairs by their difference; it is the needed finite Van der Corput inequality for finite scalar sequences. If $z_x=e(\theta x^2)$ and $\theta$ is irrational, each fixed nonzero lag correlation is a geometric sum with frequency $2\theta h$, hence is bounded independently of $L$. The geometric bound follows from $(1-z^L)/(1-z)$ with $z=e(2\theta h)\ne 1$. Dividing by $H^2{L}^2$ and letting $L$ tend to infinity gives $\mathrm{limsup}|L^{-1}\sum z_x{|}^2\le 1/H$. Letting $H$ tend to infinity proves the asserted cancellation. Rational coefficients instead give periodic sequences and therefore have convergent averages.

Choose a finite set $T$ of rational points of the circle, including zero, such that the total mass of the remaining rational atoms is below $1/(2r)$. This is possible because the rational points are countable and $\mu$ is finite. Let $q$ be a common multiple of their denominators. For $t\in T$, $e(t{q}^2{m}^2)=1$ for every integer $m$. For irrational $t$ the preceding cancellation applies to $q^{2}t$; at every other rational point the average exists by periodicity and has modulus at most one. Bounded convergence consequently givesHere the inequality takes real parts; $c$ itself is real. Thus some $b=qm>0$ has $c(b^2)>0$. By its definition as a limit of nonnegative averages, at least one actual positive $a$ has ${\sum }_j{f}_j(a)f_j(a+b^2)>0$. The same colour occurs at both points, provingThe arbitrary extension to nonpositive integers played no role in this final pair.

Assume $X$ is nonempty, as is necessary for a probability measure to exist. Give its isometry group $I_X$ the uniform metricEvery distance-preserving self-map of a compact metric space is onto. Indeed, if $x_0\notin gX$, then $\delta =d(x_0,gX)>0$, and for $n>m$ the iterates satisfy $d(g^m{x}_0,g^n{x}_0)=d(x_0,g^{n-m}{x}_0)\ge \delta$. This infinite separated set contradicts compactness. Consequently $I_X$ is a group under composition, with the identity and inverses again isometries.

The metric $D$ is invariant under both left and right multiplication: left invariance uses preservation of distances, and right invariance uses surjectivity. Inversion is an isometry for $D$, and $D(gh,g^{\prime }{h}^{\prime })\le D(g,g^{\prime })+D(h,h^{\prime })$ proves continuity of multiplication. Moreover, $I_X$ is compact. A sequence of isometries has a subsequence converging at every point of a countable dense subset of $X$, by a diagonal argument. Their common Lipschitz constant $1$ and finite epsilon-nets of $X$ make this subsequence uniformly Cauchy. Its uniform limit preserves distances and is onto by the preceding argument. Thus $I_X$ is a compact metrizable topological group. A transitive group action means that for every $x,y\in X$ some $g\in I_X$ has $gx=y$.

We construct the invariant Borel probability measure directly. The Hall marriage theorem says that a finite bipartite graph with equal-sized vertex classes $A,B$ has a perfect matching precisely when every subset $S\subseteq A$ has at least $|S|$ neighbours in $B$. Its following consequence is useful. If $A,B$ are $\epsilon$-separated subsets of maximum cardinality in a compact metric space, join points whose distance is less than $\epsilon$. If some $S\subseteq A$ had fewer than $|S|$ neighbours, replacing those neighbours in $B$ by $S$ would produce a larger $\epsilon$-separated set. This contradicts maximal cardinality. Hence there is a bijection between $A$ and $B$ moving each point by less than $\epsilon$.

Choose ${\epsilon }_n\downarrow 0$ and maximum-cardinality ${\epsilon }_n$-separated sets $A_n\subset I_X$. Such sets exist: a finite covering by balls of radius less than ${\epsilon }_n/2$ bounds every separated set's cardinality. Fix $x_0\in X$ and define the empirical probability measuresFor $f\in C(X)$ let ${\omega }_f(r)={\sup }_{d(x,y)\le r}|f(x)-f(y)|$, which tends to zero with $r$ by uniform continuity. For every $h\in I_X$, left invariance of $D$ makes $h{A}_n$ another maximum-cardinality separated set. Match it to $A_n$ as above. ThenThe allowed compactness of probability measures on a compact metric space supplies a subsequence converging to $\mu$ in the weak-star topology. Passing to that limit for each fixed $h,f$ gives $\int fd\mu =\int f\circ hd\mu$. The same subsequence works for every $h,f$, since the displayed estimate held for all of them before taking the limit.

For uniqueness, put $F_n(x)=|A_n{|}^{-1}{\sum }_{a\in A_n}f(ax)$. For $x=h{x}_0$, right invariance of $D$ lets us match $A_{n}h$ with $A_n$. Transitivity therefore gives the uniform estimate $|F_n(x)-F_n(x_0)|\le {\omega }_f({\epsilon }_n)$ for every $x\in X$. If $\nu$ is any invariant Borel probability measure, thenTwo invariant probability measures consequently differ on the integral of $f$ by at most $2{\omega }_f({\epsilon }_n)$, hence agree on every continuous function. Continuous functions determine Borel probability measures on a compact metric space, so the invariant probability measure exists and is unique.

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.