Proof of Nearly Optimal Composite Displacements on the Torus Converge to the Optimal Displacement

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Write $W=W_{\mathbb{T}}(\mu,\nu)$ and let $\iota_{\mathbb{R}}$ be the canonical map from $\mathbb{N}$ to $\mathbb{R}$. Integrals over $\mathbb{R}^{q}$ are formed in the measure space $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}),\Sigma_{n})$ and integrals over $\mathbb{R}^{d+d}$ in $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\rho_{n})$, for the measures $\rho_{n}$ of Step 1, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; by that clause a nonnegative Borel real function is integrated as a $[0,\infty]$-valued map, and for a bounded one this integral is real and equals its integral as an integrable function by claim 6(c) of Borel Measurability and Bounded Integration on a Metric Space, each probability measure being a Borel measure of total mass $1$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Composites of Borel maps are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. For a point $u$ of $\mathbb{R}^{d}$, $\lVert u\rVert^{2}=\sum_{i=1}^{d}u_{i}^{2}$ and $|u_{i}|\le\lVert u\rVert$ for every $i\in[d]$, by claims 1 and 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$.

Step 1 (The couplings $\rho_{n}$ and the excess $a_{n}$). Fix $n\in\mathbb{N}$. The pairing $(X_{n},Y_{n}):\mathbb{R}^{q}\to\mathbb{R}^{d+d}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and $\mathrm{pr}_{1}\circ(X_{n},Y_{n})=X_{n}$, $\mathrm{pr}_{2}\circ(X_{n},Y_{n})=Y_{n}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. Put $\rho_{n}=(X_{n},Y_{n})_{\#}\Sigma_{n}$, a member of $\mathcal{P}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. For $B\in\mathcal{B}(\mathbb{R}^{d})$, the defining formula of the push-forward in that clause gives $\rho_{n}(\mathrm{pr}_{1}^{-1}(B))=\Sigma_{n}((X_{n},Y_{n})^{-1}(\mathrm{pr}_{1}^{-1}(B)))=\Sigma_{n}(X_{n}^{-1}(B))=\mu(B)$, and likewise $\rho_{n}(\mathrm{pr}_{2}^{-1}(B))=\nu(B)$; so $\rho_{n}\in\Pi(\mu,\nu)$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling.

Define on $\mathbb{R}^{q}$ the functions

$$Q_{n}(w)=\lVert Z_{n}(w)\rVert^{2},\qquad P_{n}(w)=\lVert\varpi(Z_{n}(w))\rVert^{2},\qquad g_{n}(w)=Q_{n}(w)-P_{n}(w).$$

They are Borel: $\varpi$ is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz, so $Q_{n}$ and $P_{n}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (applied on the measurable space $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}))$ to $Z_{n}$ and to $\varpi\circ Z_{n}$), and $g_{n}$ by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $0\le P_{n}\le d/4$; by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal with $k=0$, $\lVert\varpi(Z_{n}(w))\rVert\le\lVert Z_{n}(w)\rVert$, and squaring (both sides being nonnegative) gives $P_{n}\le Q_{n}$, that is $g_{n}\ge0$.

Fix $w\in\mathbb{R}^{q}$ and put $m=Y_{n}(w)-X_{n}(w)-Z_{n}(w)$, which lies in $\mathbb{Z}^{d}$ by hypothesis. Then $Y_{n}(w)-X_{n}(w)=Z_{n}(w)+m$, so the periodicity $\varpi(z+m)=\varpi(z)$ of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range and The Wrapped Displacement and the Flat Torus Distance §distance give

$$d_{\mathbb{T}}\bigl(X_{n}(w),Y_{n}(w)\bigr)=\lVert\varpi(Y_{n}(w)-X_{n}(w))\rVert=\lVert\varpi(Z_{n}(w))\rVert,\qquad\varpi\bigl(Y_{n}(w)-X_{n}(w)\bigr)=\varpi\bigl(Z_{n}(w)\bigr). \tag{1}$$

The integrand of the torus cost is a bounded nonnegative Borel function on $\mathbb{R}^{d+d}$; by the change-of-variables formula and $(1)$,

$$I_{\mathbb{T}}(\rho_{n})=\int_{\mathbb{R}^{q}}d_{\mathbb{T}}\bigl(X_{n}(w),Y_{n}(w)\bigr)^{2}\,\Sigma_{n}(dw)=\int_{\mathbb{R}^{q}}P_{n}\,d\Sigma_{n}.$$

By hypothesis $\int Q_{n}\,d\Sigma_{n}<\infty$. Since $Q_{n}=g_{n}+P_{n}$ with both summands nonnegative Borel, claim 1 of Linearity and Monotonicity of the Lebesgue Integral gives $\int Q_{n}\,d\Sigma_{n}=\int g_{n}\,d\Sigma_{n}+\int P_{n}\,d\Sigma_{n}$, so both integrals on the right are real and

$$a_{n}:=\int_{\mathbb{R}^{q}}g_{n}\,d\Sigma_{n}=\int_{\mathbb{R}^{q}}Q_{n}\,d\Sigma_{n}-I_{\mathbb{T}}(\rho_{n})\ge0,\qquad I_{\mathbb{T}}(\rho_{n})\le\int_{\mathbb{R}^{q}}Q_{n}\,d\Sigma_{n}.$$

By Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance, $W$ is the nonnegative square root of the greatest lower bound of $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$, so $W^{2}\le I_{\mathbb{T}}(\rho_{n})$.

Now let $\varepsilon$ be a positive real number. By hypothesis, applied with $\varepsilon/2$, there is $N\in\mathbb{N}$ with $\int Q_{n}\,d\Sigma_{n}\le W^{2}+\varepsilon/2$ for every $n\ge N$; for such $n$,

$$W^{2}\le I_{\mathbb{T}}(\rho_{n})\le W^{2}+\tfrac{\varepsilon}{2},\qquad 0\le a_{n}\le W^{2}+\tfrac{\varepsilon}{2}-W^{2}<\varepsilon .$$

So $|I_{\mathbb{T}}(\rho_{n})-W^{2}|<\varepsilon$ and $|a_{n}-0|<\varepsilon$ for $n\ge N$, and by Limit of a Sequence of Real Numbers the sequence $(I_{\mathbb{T}}(\rho_{n}))_{n\in\mathbb{N}}$ converges to $W^{2}$ and $(a_{n})_{n\in\mathbb{N}}$ converges to $0$.

Step 2 (The wrapped part converges). As $\mu$ is absolutely continuous, the pair $(\mu,\nu)$ is uniquely mapped on the torus by McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §map. So Stability of the Optimal Map and of the Displacement on the Torus Along Couplings of Nearly Optimal Cost §displacement applies to $\mu$, $\nu$, the optimal map $T$ and the sequence $(\rho_{n})_{n\in\mathbb{N}}$ in $\Pi(\mu,\nu)$, whose costs converge to $W^{2}$ by Step 1: the sequence of integrals over $\mathbb{R}^{d+d}$ of the bounded nonnegative Borel function $u\mapsto\lVert\varpi(\mathrm{pr}_{2}(u)-\mathrm{pr}_{1}(u))-v_{T}(\mathrm{pr}_{1}(u))\rVert^{2}$ against $\rho_{n}$ converges to $0$. By the change-of-variables formula and $(1)$ this integral equals

$$b_{n}:=\int_{\mathbb{R}^{q}}f_{n}\,d\Sigma_{n},\qquad f_{n}(w)=\bigl\lVert\varpi(Z_{n}(w))-v_{T}(X_{n}(w))\bigr\rVert^{2},$$

so $(b_{n})_{n\in\mathbb{N}}$ converges to $0$. Here $f_{n}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, $v_{T}$ being Borel by Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement, and $0\le f_{n}\le d$ by the first inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, recalling $v_{T}(x)=\varpi(T(x)-x)$.

Step 3 (A gap for unwrapped points). Let $r$ be the function of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular. Let $z\in\mathbb{R}^{d}$, put $t=\varpi(z)$ and $k=z-t$, which lies in $\mathbb{Z}^{d}$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, and suppose $k\ne0$. We show

$$\lVert z\rVert^{2}-\lVert t\rVert^{2}\ge4\,r(z)^{2}. \tag{2}$$

We have $\lVert z\rVert^{2}-\lVert t\rVert^{2}=\sum_{i=1}^{d}\bigl((t_{i}+k_{i})^{2}-t_{i}^{2}\bigr)$. Fix $i\in[d]$. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $-\tfrac12\le t_{i}<\tfrac12$, so $|t_{i}|\le\tfrac12$ by claim 6 of Properties of the Absolute Value in an Ordered Field; and by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular, $r(z)\ge0$ and, $r(z)$ being the least of the numbers $\tfrac12-|t_{j}|$, $r(z)\le\tfrac12-|t_{i}|$. If $k_{i}=0$, the $i$th summand is $0$. If $k_{i}\ne0$, then $k_{i}$ is a nonzero integer, so $|k_{i}|\ge1$, and by claims 3 and 7 of Properties of the Absolute Value in an Ordered Field,

$$|t_{i}+k_{i}|\ge|k_{i}|-|t_{i}|\ge1-|t_{i}|=|t_{i}|+2\bigl(\tfrac12-|t_{i}|\bigr)\ge|t_{i}|+2r(z)\ge0 .$$

Squaring these nonnegative numbers and using $|a|^{2}=a^{2}$ (claim 4 there),

$$(t_{i}+k_{i})^{2}-t_{i}^{2}\ge\bigl(|t_{i}|+2r(z)\bigr)^{2}-t_{i}^{2}=4|t_{i}|\,r(z)+4r(z)^{2}\ge4r(z)^{2}.$$

Thus every summand is nonnegative, and since $k\ne0$ some $k_{i}$ is nonzero, whose summand is at least $4r(z)^{2}$; this proves $(2)$.

Step 4 (The regularity radius along $T$). By McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §potential, applied to the optimal map $T$, there is $D\in\mathcal{B}(\mathbb{R}^{d})$ with $\mu(D)=1$ such that $T(x)-x$ is regular for every $x\in D$. Put $\rho(x)=r(T(x)-x)$ for $x\in\mathbb{R}^{d}$. The map $\rho$ is Borel: $r$ is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular, and $T-\mathrm{id}$ is Borel as recorded in Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular, $\rho(x)>0$ for $x\in D$. Since $v_{T}(x)=\varpi(T(x)-x)$ by Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement, $\rho(x)$ is the least of the numbers $\tfrac12-|v_{T}(x)_{j}|$, $j\in[d]$.

We record: for $z,x\in\mathbb{R}^{d}$, with $t=\varpi(z)$ and $e=t-v_{T}(x)$,

$$r(z)\ge\rho(x)-\lVert e\rVert. \tag{3}$$

Indeed, for each $i\in[d]$, claim 5 of Properties of the Absolute Value in an Ordered Field and the coordinate bound give $|t_{i}|\le|v_{T}(x)_{i}|+|e_{i}|\le|v_{T}(x)_{i}|+\lVert e\rVert$, hence $\tfrac12-|t_{i}|\ge\tfrac12-|v_{T}(x)_{i}|-\lVert e\rVert\ge\rho(x)-\lVert e\rVert$; and $r(z)$ is one of the numbers $\tfrac12-|t_{i}|$.

For $m\in\mathbb{N}$ put $\eta_{m}=\iota_{\mathbb{R}}(m)^{-1}$, a positive real by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and

$$A_{m}=\{x\in D:2\eta_{m}<\rho(x)\},$$

which belongs to $\mathcal{B}(\mathbb{R}^{d})$ because $\{x:2\eta_{m}<\rho(x)\}$ does, by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable for the measurable space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$.

Step 5 (A pointwise bound). Fix $m,n\in\mathbb{N}$, write $\eta=\eta_{m}$, $A=A_{m}$, $C=4+2d\,\eta^{-2}$, and put $B=X_{n}^{-1}(\mathbb{R}^{d}\setminus A)\in\mathcal{B}(\mathbb{R}^{q})$ ($X_{n}$ being Borel) and

$$F_{n}(w)=\bigl\lVert Z_{n}(w)-v_{T}(X_{n}(w))\bigr\rVert^{2}\qquad(w\in\mathbb{R}^{q}),$$

a Borel function by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions. We show that for every $w\in\mathbb{R}^{q}$

$$F_{n}(w)\le C\,g_{n}(w)+C\,f_{n}(w)+2d\,\mathbf{1}_{B}(w). \tag{4}$$

Fix $w$ and write $z=Z_{n}(w)$, $x=X_{n}(w)$, $t=\varpi(z)$, $k=z-t\in\mathbb{Z}^{d}$ and $e=t-v_{T}(x)$, so that $g_{n}(w)=\lVert z\rVert^{2}-\lVert t\rVert^{2}$, $f_{n}(w)=\lVert e\rVert^{2}$ and $z-v_{T}(x)=k+e$. By the first inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to $k$ and $-e$ (note $\lVert-e\rVert=\lVert e\rVert$ by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$),

$$F_{n}(w)\le2\lVert k\rVert^{2}+2f_{n}(w).$$

All terms on the right of $(4)$ are nonnegative, $g_{n}\ge0$ by Step 1, and $C\ge2$. If $k=0$, then $F_{n}(w)\le2f_{n}(w)\le C f_{n}(w)$ and $(4)$ holds. Let $k\ne0$. The same inequality, applied to $z$ and $t$, together with $\lVert t\rVert^{2}\le d/4$ from The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, gives $\lVert k\rVert^{2}\le2\lVert z\rVert^{2}+2\lVert t\rVert^{2}=2g_{n}(w)+4\lVert t\rVert^{2}\le2g_{n}(w)+d$, hence

$$F_{n}(w)\le4g_{n}(w)+2f_{n}(w)+2d .$$

We bound $1$ by one of three nonnegative quantities. (i) If $x\notin A$, then $w\in B$ and $1=\mathbf{1}_{B}(w)$. (ii) If $x\in A$ and $\eta\le\lVert e\rVert$, then squaring gives $\eta^{2}\le f_{n}(w)$, so $1\le\eta^{-2}f_{n}(w)$. (iii) If $x\in A$ and $\lVert e\rVert<\eta$, then $2\eta<\rho(x)$, so $(3)$ gives $r(z)>2\eta-\eta=\eta>0$; as $k\ne0$, $(2)$ gives $g_{n}(w)\ge4r(z)^{2}\ge4\eta^{2}$, so $1\le\tfrac14\eta^{-2}g_{n}(w)$. In every case $1\le\mathbf{1}_{B}(w)+\eta^{-2}f_{n}(w)+\tfrac14\eta^{-2}g_{n}(w)$, and therefore

$$F_{n}(w)\le\bigl(4+\tfrac12d\,\eta^{-2}\bigr)g_{n}(w)+\bigl(2+2d\,\eta^{-2}\bigr)f_{n}(w)+2d\,\mathbf{1}_{B}(w),$$

which implies $(4)$ because both coefficients are at most $C$ and $g_{n}(w),f_{n}(w)\ge0$.

The right side of $(4)$ is a sum of nonnegative Borel functions (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions for the indicator). Integrating $(4)$ against $\Sigma_{n}$ with claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and using The Integral of an Indicator Function is the Measure of the Set together with $\Sigma_{n}(X_{n}^{-1}(\mathbb{R}^{d}\setminus A))=((X_{n})_{\#}\Sigma_{n})(\mathbb{R}^{d}\setminus A)=\mu(\mathbb{R}^{d}\setminus A)$ from Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, we obtain for all $m,n\in\mathbb{N}$

$$0\le c_{n}:=\int_{\mathbb{R}^{q}}\bigl\lVert Z_{n}-v_{T}\circ X_{n}\bigr\rVert^{2}\,d\Sigma_{n}\le\bigl(4+2d\,\eta_{m}^{-2}\bigr)(a_{n}+b_{n})+2d\,\mu(\mathbb{R}^{d}\setminus A_{m}). \tag{5}$$

In particular each $c_{n}$ is a real number.

Step 6 (The exceptional set is small). By claims 1 and 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $0<\iota_{\mathbb{R}}(m)<\iota_{\mathbb{R}}(m)+1=\iota_{\mathbb{R}}(m+1)$, so $\eta_{m+1}\le\eta_{m}$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal, applied with $a=\iota_{\mathbb{R}}(m)$ and $b=\iota_{\mathbb{R}}(m+1)$; therefore $A_{m}\subseteq A_{m+1}$. Every $A_{m}$ is contained in $D$; conversely, for $x\in D$ one has $\rho(x)/2>0$ (Step 4), and claim 3 of The Archimedean Property of the Real Numbers gives $l\in\mathbb{N}$ with $0<\iota_{\mathbb{R}}(l)^{-1}<\rho(x)/2$, so $x\in A_{l}$. Hence $\bigcup_{m}A_{m}=D$. Each $\mu(A_{m})$ is a real number at most $\mu(\mathbb{R}^{d})=1$, by claim 2 of Basic Properties of a Measure, so claim 5 there (continuity from below) shows that $(\mu(A_{m}))_{m}$ converges to $\mu(D)=1$; and $\mu(\mathbb{R}^{d}\setminus A_{m})=1-\mu(A_{m})$ by claim 3 there. Hence $(\mu(\mathbb{R}^{d}\setminus A_{m}))_{m\in\mathbb{N}}$ converges to $0$.

Step 7 (Conclusion). Let $\varepsilon$ be a positive real number. By Step 6 choose $m\in\mathbb{N}$ with $\mu(\mathbb{R}^{d}\setminus A_{m})<\varepsilon/(4d)$, and put $C=4+2d\,\eta_{m}^{-2}$, a positive real. By Steps 1 and 2 choose $N_{1},N_{2}\in\mathbb{N}$ with $a_{n}<\varepsilon/(4C)$ for every $n\ge N_{1}$ and $b_{n}<\varepsilon/(4C)$ for every $n\ge N_{2}$, and let $N$ be the larger of $N_{1}$ and $N_{2}$. The order of choice is $\varepsilon$, then $m$, then $N_{1}$ and $N_{2}$. For every $n\ge N$, $(5)$ gives

$$|c_{n}-0|=c_{n}<C\Bigl(\frac{\varepsilon}{4C}+\frac{\varepsilon}{4C}\Bigr)+2d\cdot\frac{\varepsilon}{4d}=\varepsilon .$$

By Limit of a Sequence of Real Numbers the sequence $(c_{n})_{n\in\mathbb{N}}$ converges to $0$, which is claim 1.