Background and significance

Erdős initiated the planar distinct-distance and repeated-distance problems in 1946 (Erdős 1946). The global distinct-distance problem asks for the minimum size of the union \(\bigcup_{x\in P}D_x(P)\). The square lattice gives sets for which this union has size at most a constant multiple of \(n/\sqrt{\log n}\) (Erdős 1946, Theorem 1). Guth and Katz proved that every planar \(n\)-point set determines at least a constant multiple of \(n/\log n\) distances (Guth and Katz 2015, Theorem 1.1). Their conclusion concerns the union; the pinned question requires the distances to have a common source.

The development of pinned bounds combined incidence geometry with increasingly effective counting and entropy estimates. Solymosi and Tóth proved a pinned lower bound of order \(n^{6/7}\) (Solymosi and Tóth 2001, Theorem 1). Tardos improved this exponent using entropy inequalities for patterns of sums and differences (Tardos 2003). Katz and Tardos refined that method to obtain every exponent below \[\frac{48-14e}{55-16e}=0.864137\ldots\] (Katz and Tardos 2004). These results guarantee one pin in every configuration. The sharper pinned conjecture asks for order \(n/\sqrt{\log n}\) distances at a pin; see (Lund and Petridis 2020, Introduction). Lund and Petridis also explain the distinction between the real planar problem and pinned-distance theorems over other fields.

Theorem 1 controls the multiplicity of a distance chosen by sampling a distinct ordered pair. Corollary 2 makes its almost-every-pin consequence explicit and resolves the weak pinned conjecture. An exceptional set is necessary: if \(P\) consists of points on a circle and its center, the center sees just one nonzero distance.

The proof in outline

Assuming that Theorem 1 fails, we first choose a sequence with nearly maximal density of pairs in large distance fibers. Section 2 extracts a directed graph of these pairs while controlling how much of a fiber can lie in a small subset. Transfer for real closed fields preserves every distance equality and disequality and puts each finite configuration in a number field (Kuhlmann 2009, Theorems 3.1 and 4.1). Neither the field nor its degree is fixed.

The proof then studies distance multiplicities through two coordinate maps. For \(x=(u,v)\), set \(Z_1(x)=u+iv\) and \(Z_2(x)=u-iv\). Squared distance factors as \[\lVert x-y\rVert_2^2 =(Z_1(y)-Z_1(x))(Z_2(y)-Z_2(x)).\] Thus on a distance fiber at \(x\), the second coordinate is a fractional linear function of the first. The normalized product formula balances the logarithms of nonzero coordinate differences across all absolute values of the number field (Milne 2020, Theorem 7.15). At other complex embeddings the two maps need not be conjugates; the algebraic factorization remains valid.

We turn this balance into an identity for finite nested partitions. At each absolute value, points are grouped according to closeness. Randomly shifted and rotated square grids provide nested partitions at the complex embeddings. At any level, a cell containing more than half the points, called a giant, is replaced by its complement, with multiplicities retained. The resulting overlap of two distinct points, summed over all levels and absolute values, is the sum of a function of the first point and a function of the second. This additive identity is the main use of arithmetic in the proof.

Randomly shifted square dissections appear in Arora’s Euclidean approximation algorithms (Arora 1998, sec. 2.1). Nested partitions and their associated trees also occur in probabilistic metric approximation, as in the work of Fakcharoenphol, Rao and Talwar (Fakcharoenphol et al. 2003). Their approximation controls multiplicative distortion. Here the required grid estimate has a pair-independent mean additive error in logarithmic depth; this specific property permits the product-formula cancellation.

We measure the total scale by the integrated distinct-pair overlaps and the average lengths of levels spent outside giant cells. The next step is to compare the uniform distribution on a fiber with the uniform distribution on the whole configuration. The extremal subset estimate controls cells that contain a source and a substantial fraction of its fiber. For an actual partition cell containing only a small fraction, the other coordinate supplies the missing control: without grid errors, its source cell at the complementary level contains the rest of the fiber. Together these estimates make the small-cell contribution negligible relative to one plus the total scale (Section 4).

We then compare three laws on distinct ordered pairs: the uniform law within a fiber, a law with one fiber-uniform and one ambient-uniform marginal, and the uniform law on the whole configuration. The first minus twice the second plus the third has zero total mass and zero sum of its marginals. Its integral against the additive overlap kernel therefore vanishes. Expanding the cell probabilities gives a squared discrepancy between fiber mass and uniform mass, together with explicit finite-population corrections. Large support sizes and the small-cell estimate control these corrections. Section 5 establishes the resulting variance bound, both for fibers and for the sources of edges ending at a target. The use of distinct pairs is essential: singleton cells persist through an infinite range of levels, over which even a small diagonal error could not be integrated.

After passing to a subsequence, this nonnegative scale either grows without bound or stays bounded. In the first case, we retain cells whose mass and complementary mass are both bounded below. They give weighted finite rooted trees with total length comparable to the scale. The fiber relation and the fiber variance bound make the source-to-target distance in one tree close to the root-to-target distance in the other. The latter depends only on the target. Transport on the trees, using the incoming-law variance bound, lets us replace two sources arriving at the same target by independent uniform points. After replacing the target law as well, the expected absolute difference of their distances to that target would be negligible relative to the total overlap scale. An elementary geometric bound for trees contradicts this conclusion (Section 6).

If the overlap scale stays bounded, one complex embedding can be normalized so that the two coordinate distributions have non-atomic limits. The fiber and incoming-neighbor estimates yield a family of Möbius maps carrying one limiting coordinate law to the other. The first source coordinate is independent of the entire target pair; the second source coordinate need not be. To remove the latter dependence, fix a target pair and sample a point from the first limiting coordinate law. Its image under the fiber map has the second limiting law. The fiber equation writes the negative logarithm of the distance from that image to the second target coordinate as a difference of logarithmic distances from the sampled point to the first source and target coordinates, up to an additive constant. Independence lets the first source coordinate approach the first target coordinate. This difference then collapses in probability under the fixed sampling law, whereas the image’s logarithmic target-distance law has positive spread even after additive constants are discarded. Section 7 makes this contradiction precise.

The tree-transport formula is a standard sum of the imbalances across tree edges (Evans and Matsen 2012, sec. 2, Equation (5)); we prove its finite form and the conditional comparisons needed here. Weak compactness and Portmanteau provide subsequential probability laws (Gaans 2003, Proposition 5.3 and Theorems 4.2, 3.2). The local arguments must additionally establish non-atomicity, independence of the source parameter, and preservation of the fiber map. These properties connect the standard tools to the additive overlap identity and its exact variance consequence.

Section 2 prepares the extremal graph and its sampling laws. Section 3 constructs the partitions and proves the additive identity. Sections 4 and 5 establish the overlap and variance estimates. Sections 6 and 7 exclude the two scale regimes and complete the proof of Theorem 1.

Two coordinates at every place

Fix one of the algebraic configurations supplied by Proposition 4. Take a number field \(K\subset\mathbb C\) containing all its coordinates and \(\mathrm i=\sqrt{-1}\). For a point \(x=(u,b)\) set \[Z_1(x)=u+\mathrm i b,\qquad Z_2(x)=u-\mathrm i b.\] Both maps are injective, because \(u,b\) are real in the given embedding. Their differences remain nonzero under every field embedding of \(K\). At another complex embedding the two coordinates need not be complex conjugates; their injectivity and the algebraic factorization below are the properties we use. For distinct \(x,y\) the squared distance factors in \(K\) as \[ t(x,y)=(Z_1(y)-Z_1(x))(Z_2(y)-Z_2(x)). \tag{6}\]

Let \(v\) run over the places of \(K\). At a finite place above a prime \(p\), use the absolute value extending \(|p|_p=p^{-1}\) and put \(\sigma_v=[K_v:\mathbb Q_p]/[K:\mathbb Q]\). All archimedean places are complex, since \(\mathrm i\in K\); there use ordinary complex modulus and \(\sigma_v=2/[K:\mathbb Q]\). The normalized product formula (Milne 2020, Theorems 7.14–7.15 and Lemma 8.6) reads \[ \sum_v\sigma_v\log|z|_v=0\quad(z\in K\setminus\{0\}), \qquad \sum_{v\mid\infty}\sigma_v=1. \tag{7}\] The weights convert the norm-based finite absolute values and squared complex moduli in that reference to the absolute values used here, after division by \([K:\mathbb Q]\). Here and below logarithms are natural. For a fixed configuration only finitely many finite places make any coordinate difference \(Z_j(x)-Z_j(y)\) a nonunit: each nonzero element of \(K\) has nonzero valuation at only finitely many prime ideals, and there are finitely many differences.

At a place \(v\), define the raw depths \[i_j(x,y)=-\log|Z_j(x)-Z_j(y)|_v\quad(x\ne y),\qquad j=1,2.\] The place is suppressed in this notation. On a fiber \(F_e\) with source \(x_e\) and squared distance \(t_e\), Equation (6) gives \[ i_1(x_e,y)+i_2(x_e,y)=h_e, \qquad h_e=-\log|t_e|_v,\qquad y\in F_e. \tag{8}\] The number \(h_e\) is finite and independent of the choice of \(y\) in the fiber.

Nested grids and their error

At a finite place, put \(d_j(x,y)=i_j(x,y)\) for distinct points. The ultrametric inequality makes \(d_j(x,y)\ge r\) an equivalence relation. At a complex place we obtain the same nesting property from square grids, with a controlled error in the depth.

Independently of all vertex and fiber sampling, choose independently a uniformly random orientation and \(\xi\) uniform on \([0,\log2)\). The grid side lengths are \(l_m=e^\xi2^m\), \(m\in\mathbb Z\). Choose a shift uniform modulo \(l_0\) in each oriented axis. To obtain the shift modulo \(l_{m+1}\) from that modulo \(l_m\), for \(m\ge0\), add either \(0\) or \(l_m\) with equal probabilities, independently at every step and in each axis. For \(m<0\) reduce the initial shift modulo \(l_m\). Every shift is uniform modulo its side length, and the resulting half-open square grids are nested. Use such a grid family for each coordinate \(j\). Randomly shifted recursive square dissections also appear in Arora’s Euclidean approximation algorithms (Arora 1998, sec. 2.1).

For distinct points, let \(d_j(x,y)\) be minus the logarithm of the smallest grid side at which their images share a cell. In both the finite and complex cases put \(d_j(x,x)=+\infty\) and \(\Delta_j(x,y)=d_j(x,y)-i_j(x,y)\) for \(x\ne y\).

The three depth quantities have distinct roles:

Removing the giant cell

Discard the null event on which any distinct-pair depth is not finite. For either type of place let \(\Pi_j(r)\) be the partition with pair-sharing criterion \(d_j(x,y)\ge r\), for \(r\in\mathbb R\). These partitions are nested as \(r\) increases. Call a cell a giant if its size exceeds \(n/2\), and denote the unique such cell, when it exists, by \(G_j(r)\). Replace that cell by its complement and keep all other cells. The resulting family \(\mathcal C_j(r)\) is a multiset: if the complement is itself one other cell, it occurs twice. Every member \(C\) has \(\alpha(C)\le1/2\). Figure 1 illustrates why multiplicities matter.

This replacement removes the overlap of all pairs at very coarse scales. Its cost is a function of each endpoint separately, which will disappear when pair laws with the same combined marginals are compared. Define \[\begin{align*} M_j&=\sup\{r:G_j(r)\text{ exists}\},\\ S_j(x)&=\int_{\{r:G_j(r)\text{ exists}\}} \mathbf 1_{\{x\notin G_j(r)\}}\,dr,\\ L_j(x,y)&=\int_\mathbb R\sum_{C\in\mathcal C_j(r)} \mathbf 1_C(x)\mathbf 1_C(y)\,dr\qquad(x\ne y). \end{align*}\]

The empty complement of \(P\) and singleton occurrences contribute zero to every distinct-pair overlap. We do not define \(L_j(x,x)\): singleton cells persist to arbitrarily fine scales, so its integral would be infinite.

The exact overlap identity

For a nonnegative or integrable quantity depending on the place and grids, write \[\left\langle H\right\rangle=\sum_v\sigma_v\mathbb E_{\mathrm{grids}}H, \qquad \mathcal I(g)=\left\langle \sum_{j=1}^2\int_\mathbb R \sum_{C\in\mathcal C_j(r)}g\,dr\right\rangle.\] There is no grid randomness at finite places. The integrand \(g\) may also depend on the coordinate, level, and cell occurrence. All sums over \(\mathcal C_j(r)\) retain multiplicity. These notations do not assert finiteness for an arbitrary \(g\); finiteness will be established for each expression used.

Set \[S^*(x)=\left\langle S_1(x)+S_2(x)\right\rangle,\qquad L^*(x,y)=\left\langle L_1(x,y)+L_2(x,y)\right\rangle\quad(x\ne y).\] Lemma 6 and the finite set of nonunit places show that these quantities, and \(\left\langle M_1+M_2\right\rangle\), are finite. For each distinct pair, the product formula and Lemma 5 give \[\left\langle d_1(x,y)+d_2(x,y)\right\rangle=2\kappa.\] Taking expectations in (10) is therefore legitimate and yields the promised additive identity \[ L^*(x,y)=b_*+S^*(x)+S^*(y)\quad(x\ne y), \qquad b_*=2\kappa-\left\langle M_1+M_2\right\rangle. \tag{11}\] In particular, any signed law on distinct pairs with zero total mass and zero sum of its two marginals has zero integral against \(L^*\). This consequence is exact; no error from sampling a diagonal pair is involved.

The three nonnegative finite scales used below are \[ W_2=\mathcal I\bigl(\pi(C^2)\bigr)=\mathbb E_\pi L^*(x,y),\qquad A=\mathbb E_\alpha S^*(x),\qquad W=W_2+A. \tag{12}\] Thus \(W_2\) measures distinct-pair overlap after the giant is replaced, while \(A\) measures the average time spent outside the giant. Equation (11) relates these scales to the graph and fiber laws. Every absolute grid-error bound remains uniform in \(K\), since its archimedean weights sum to one.

Extracting cells of positive mass

We first claim that \[ W_2\ge c_1 W \tag{26}\] for some fixed \(c_1>0\) and all sufficiently large \(n\). At a fixed place and choice of grids, abbreviate \(S=S_1+S_2\) and \(L=L_1+L_2\). Choose a fiber \(e\) with its edge-induced weight and then choose \(Y,Y'\) independently with law \(\mu_e\). Equation (10) and the constant raw depth sum on the fiber give \[S(Y)=M_1+M_2-S(x_e)-h_e+L(x_e,Y) -\Delta_1(x_e,Y)-\Delta_2(x_e,Y).\] All source–neighbor pairs are distinct. Thus the first-moment bound from (9), the normalization of the archimedean weights, and (13) imply \[\left\langle \mathbb E|S(Y)-S(Y')|\right\rangle\le 2\left\langle \mathbb E_q L(x,y)\right\rangle+O(1) =O(W_2)+O(1).\] To compare this with \(A\), we use the pointwise bound \[ S^*(x)\ge A-2W_2\qquad(n>2). \tag{27}\] Indeed, the average of \(L^*(x,y)\) over the \(n-1\) labels \(y\ne x\) equals \[W_2+\frac{n-2}{n-1}\bigl(S^*(x)-A\bigr),\] and is nonnegative. Hence \(\left\langle \mathbb ES(Y)\right\rangle\ge A-2W_2\). Consequently \[ \left\langle \mathbb E\min\{S(Y),S(Y')\}\right\rangle\ge A-O(W_2)-O(1). \tag{28}\]

For two nonnegative lists \((a_1,a_2)\) and \((b_1,b_2)\), \[\min\{a_1+a_2,b_1+b_2\}\le\sum_{i,j=1}^2\min\{a_i,b_j\}.\] Indeed, if the first sum is smaller, then \(\sum_j\min\{a_i,b_j\}\ge a_i\) for each \(i\). Moreover, nesting of the giants gives, for almost every \(u>0\), \[\{z:S_i(z)>u\}=P\setminus G_i(M_i-u).\] A vertex which leaves the giant at level \(r_z<M_i\) has \(S_i(z)=M_i-r_z\); one which never leaves has \(S_i(z)=0\). This proves the displayed identity, with only finitely many possible threshold ties. Conditional independence of \(Y,Y'\) and \(ab\le(a^2+b^2)/2\) now give \[ \left\langle \mathbb E\min\{S(Y),S(Y')\}\right\rangle \le 2\left\langle \sum_{i=1}^2\mathbb E_e\int_{r<M_i} \mu_e(P\setminus G_i(r))^2\,dr\right\rangle. \tag{29}\] To bound the right side, write \(p=\mu_e(C)\) and \(a=\alpha(C)\) for a giant-complement occurrence. If \(|C|>n^{1-c}\), use \(p^2\le2(p-a)^2+2a^2\), the variance estimate (19), and \(a^2\le2\pi(C^2)\). For smaller complements with \(p\ge2/k_e\), use the second term of (19). The remaining terms satisfy \(p^2\le2n^{-\gamma}p\). Since \(\mathbb E_e p=q_Y(C)\le B_0a\), their integral is \(O(n^{-\gamma}A)\). In particular, this last error is charged to \(A\), the finite integral of the complement masses. We obtain \[\left\langle \mathbb E\min\{S(Y),S(Y')\}\right\rangle \le O(W_2)+o(W)+O(n^{-\gamma}A).\] Together with (28), this proves \(A\le O(W_2)+o(W)+O(1)\), and hence (26). Equation (13) also gives \(\mathcal I(q(C^2))\gg W\).

We next restrict the masses of the cells we use. For a fixed \(0<\delta<1/4\), split the contribution of \(a\le\delta\) according to \(p<2\delta\) or \(p\ge2\delta\). In the latter case \(p\le p^2/(2\delta)\) and \(p-a\ge p/2\). On large cells this is controlled by the first variance term; on small cells, \(p\ge2\delta\ge2/k_e\) eventually, so the second variance term applies. Therefore \[\mathcal I\bigl(\mathbf 1_{\{a\le\delta\}}q(C^2)\bigr) \le T(2\delta)+o_\delta(W) \le O(\delta^{1/3}W)+o_\delta(W).\] For the last inequality apply Lemma 7 with \(\lambda=2\delta\), \(\tau=\lambda^{1/3}\) and any fixed \(H\ge1\). The terms depending only on \(\delta,H\) are \(o_\delta(W)\). The implicit constant in \(O(\delta^{1/3}W)\) is independent of \(\delta\). We may thus choose \(\delta\) sufficiently small once and for all, and then let \(n\to\infty\), to obtain \[ \mathcal I(\mathbf 1_{\{a>\delta\}})\gg W, \qquad \mathcal I(\mathbf 1_{\{a\ge\delta/4\}})\ll_\delta W. \tag{30}\] The first inequality uses \(q(C^2)\le1\). For the second, when \(n\) is large, \(a\ge\delta/4\) implies \(\pi(C^2)\ge a^2/2\ge\delta^2/32\). All cutoffs below use this fixed \(\delta\); in particular, a bound \(O_\delta(1)\) is \(o(W)\) along the present subsequence.

Choose a Lipschitz function \(w:[0,1]\to[0,1]\) satisfying \[w(u)=w(1-u),\qquad w=0\text{ on }[0,\delta/2]\cup[1-\delta/2,1],\qquad w=1\text{ on }[\delta,1-\delta].\] For the original partitions, define \[\ell_j=\int_{\mathbb R}\sum_{B\in\Pi_j(r)}w(\alpha(B))\,dr.\] Replacing a giant by its complement preserves every summand by symmetry, with occurrences still counted separately. Thus \[ W\ll\left\langle \ell_1+\ell_2\right\rangle\ll_\delta W. \tag{31}\]

For either family of measures in Lemma 8, we shall use \[ \left\langle \mathbb E_{\mathcal F}\sum_{j=1}^2\int_{\mathbb R} \sum_{B\in\Pi_j(r)} \left[|w(\nu(B))-w(\alpha(B))| +w(\alpha(B))|\nu(B)-\alpha(B)|\right]dr\right\rangle=o(W). \tag{32}\] The first term lets us replace cut weights defined by the fiber laws with those defined by the uniform law. The second controls the weighted mass imbalance that will appear in the transport comparison. Here the family of incoming laws retains only targets with at least \(n^\gamma\) incoming neighbors, as in Section 5. To prove (32), note first that its integrand is unchanged under \((p,a)\mapsto(1-p,1-a)\). Work therefore with the transformed cells. On \(a\ge\delta/4\), all cells are large for large \(n\); the Lipschitz bound, Cauchy–Schwarz, (30), and (19) give \(o(W)\). On \(a<\delta/4\), the second summand vanishes, and the first can be nonzero only if \(p>\delta/2\). For large cells, \(1\le16\delta^{-2}(p-a)^2\); for small cells, \(1\le4\delta^{-2}p^2\) and \(p\ge2/k\) eventually. The same variance estimate again gives \(o(W)\).

Distances obtained from the hierarchy

We have retained a positive total amount of cells, and the masses of those cells are close to their \(\alpha\)-masses in the fiber average. We now use these cells as cuts defining a distance. For \(\nu=\alpha\) or \(\mu_e\), put \[D_j^\nu(a_0,a_1)=\int_{\mathbb R}\sum_{B\in\Pi_j(r)} |\mathbf 1_B(a_0)-\mathbf 1_B(a_1)|\,w(\nu(B))\,dr.\] We also allow a formal endpoint \(\infty\), setting \(\mathbf 1_B(\infty)=0\) for every cell. These quantities are finite for all sufficiently large \(n\). On the initial ray the only cell is \(P\), of weight \(w(1)=0\); on the final ray all cells are singletons, of \(\nu\)-mass either \(0\), \(1/n\), or \(1/k_e\), so their weights vanish too.

For distinct \(a_0,a_1\in P\) we have \[ D_j^\nu(a_0,a_1)=\int_{\mathbb R} w\bigl(\nu\{z:d_j(a_1,z)-d_j(a_0,z)\ge u\}\bigr)\,du. \tag{33}\] Indeed, set \(d_0=d_j(a_0,a_1)\). Nestedness implies that if \(d_j(a_1,z)>d_0\), then \(d_j(a_0,z)=d_0\), and conversely with the indices exchanged. Otherwise the two depths to \(z\) agree. For \(u>0\) the event in (33) is the \(a_1\)-cell at level \(d_0+u\). For \(u<0\) its complement is the \(a_0\)-cell at level \(d_0-u\), up to strict versus weak threshold ties. There are only finitely many finite depths, so those ties have zero Lebesgue measure. Splitting the integral at zero and using \(w(t)=w(1-t)\) gives exactly the two cell contributions above their common depth \(d_0\). The depth difference is \(+\infty\) at \(z=a_1\) and \(-\infty\) at \(z=a_0\); no difference \(\infty-\infty\) occurs. Directly from the cell definition, we also have \[ D_j^\nu(\infty,y)=\int_{\mathbb R} w\bigl(\nu\{z:d_j(y,z)\ge u\}\bigr)\,du. \tag{34}\]

For an edge \((x,y)\) in fiber \(e\), put \(j'=3-j\). The fiber equation implies, for \(z\in F_e\), \[Z_{j'}(z)=Z_{j'}(x)+\frac{t_e}{Z_j(z)-Z_j(x)}.\] Consequently, for \(z\ne y\) in this fiber, \[Z_{j'}(z)-Z_{j'}(y) =\frac{t_e\bigl(Z_j(y)-Z_j(z)\bigr)} {(Z_j(z)-Z_j(x))(Z_j(y)-Z_j(x))},\] and hence \[ i_{j'}(y,z)=i_j(y,z)-i_j(x,z)+h_e-i_j(x,y). \tag{35}\] This identity converts one of the projected distances into a root distance in the other coordinate. Here are the details, including the diagonal atom. Under \(z\sim\mu_e\), compare \[U(z)=d_{j'}(y,z),\qquad V(z)=d_j(y,z)-d_j(x,z)+h_e-i_j(x,y).\] For \(z\ne y\), Equation (35) gives \[|U(z)-V(z)|\le |\Delta_{j'}(y,z)|+|\Delta_j(y,z)|+|\Delta_j(x,z)|.\] At \(z=y\), both variables are \(+\infty\), so their indicators at every finite threshold agree; \(z=x\) has zero \(\mu_e\)-mass. Tonelli’s theorem therefore gives the precise bound \[\int_{\mathbb R}|\Pr(U\ge u)-\Pr(V\ge u)|\,du \le \sum_{z\in F_e\setminus\{y\}}\mu_e(z)|U(z)-V(z)|.\] Translation of the threshold by \(h_e-i_j(x,y)\) does not change the integral of the weighted tail. Apply the Lipschitz bound for \(w\), (33), and (34). After averaging over \(e\), \(y\sim\mu_e\), grids and places, we obtain \[\left\langle \mathbb E_q |D_j^{\mu_e}(x,y)-D_{j'}^{\mu_e}(\infty,y)|\right\rangle=O_\delta(1).\] The error estimate is valid for these sampling laws because (9) bounds the grid first moment uniformly for every distinct pair. All vertex laws are independent of the grids; nonarchimedean errors vanish and the archimedean weights sum to one.

Changing \(\mu_e\) to \(\alpha\) changes any one of its cut distances by at most \(\int\sum_B|w(\mu_e(B))-w(\alpha(B))|\,dr\), independently of the endpoints. Thus (32), with \(D_j=D_j^\alpha\), yields \[ \left\langle \mathbb E_q|D_j(x,y)-D_{j'}(\infty,y)|\right\rangle=o(W). \tag{36}\] The remaining task is to replace the edge law in this equation by independent uniform vertices and then exploit the geometry of the cut distances.

Transport to independent uniform vertices

Each \(D_j\) is the path distance on a finite weighted tree, allowing zero-length edges. To see this, take the distinct cells of the nested partitions as vertices, root the inclusion tree at \(P\), and connect each proper cell \(B\) to its least strict supercell. Give this edge length \[l_j(B)=\int_{\mathbb R}\mathbf 1_{\{B\in\Pi_j(r)\}}w(\alpha(B))\,dr.\] The points of \(P\) are singleton leaves. An edge is crossed by their connecting path exactly when its cell separates them, proving the cut formula for \(D_j\). The total edge length is exactly \(\ell_j\). The formal endpoint is represented by the root, since the root-cell weight is zero.

For probabilities \(\rho,\alpha\) on the leaves, let \(\mathsf T_j(\rho,\alpha)\) be the least expected \(D_j\)-distance among couplings of the two laws. The tree transport formula is \[ \mathsf T_j(\rho,\alpha) =\sum_{B\ne P}l_j(B)|\rho(B)-\alpha(B)| =\int_{\mathbb R}\sum_{B\in\Pi_j(r)} w(\alpha(B))|\rho(B)-\alpha(B)|\,dr. \tag{37}\] This standard formula is the finite-tree form of (Evans and Matsen 2012, sec. 2, Equation (5)). It has a short direct proof. Across the edge above \(B\), any coupling must move at least \(|\rho(B)-\alpha(B)|\) units of mass. Conversely, first match common mass at each leaf. Work upward, matching surplus mass from child subtrees to deficits in other child subtrees, and send only the remaining imbalance toward the parent. Across each edge the total mass moved is precisely its absolute imbalance; the root imbalance is zero. Decomposing these finitely many transfers into leaf-to-leaf paths gives a coupling attaining the displayed cost. Zero-length edges do not affect the argument.

By \(q_Y=\mathbb E_e\mu_e\), convexity and (32) give \[ \left\langle \mathsf T_j(q_Y,\alpha)\right\rangle=o(W),\qquad \left\langle \mathbb E_{q_Y}\mathsf T_j(\rho_y,\alpha)\right\rangle=o(W). \tag{38}\] For the second estimate, first apply (32) to targets with at least \(n^\gamma\) incoming neighbors. The omitted \(q_Y\)-mass is \(O(n^{\gamma-1})=o(1)\), independent of the hierarchy. Every transport cost is at most \(\ell_j\), so its omitted contribution is \(o(1)\left\langle \ell_j\right\rangle=o(W)\) by (31).

The conditional source law in Equation (36), given the target \(y\), is \(\rho_y\). We keep this conditioning while coupling the sources to the uniform law. Fix a hierarchy for a moment and put \(F(x,x',y)=|D_j(x,y)-D_j(x',y)|\). If \(Y\sim q_Y\) and \(X,X'\) are independent with conditional law \(\rho_Y\), then \[\mathbb EF(X,X',Y) \le2\mathbb E_q|D_j(x,y)-D_{j'}(\infty,y)|.\] For each \(y\), use two independent copies of an optimal coupling of \(\rho_y\) to \(\alpha\), replacing \(X,X'\) by \(U,U'\). Conditional on \(y\), the new points are independent with law \(\alpha\); since that law does not depend on \(y\), they are also independent of \(Y\). The triangle inequality bounds the change in \(F\) by the sum of the two transport distances. Next optimally couple \(Y\sim q_Y\) to \(V\sim\alpha\), independently of \(U,U'\). Moving the last argument of \(F\) costs at most twice its transport distance. We have proved the pointwise inequality \[\begin{align*} \mathbb E_{\alpha^{\otimes3}}F &\le 2\mathbb E_q|D_j(x,y)-D_{j'}(\infty,y)|\\ &\quad+2\mathbb E_{q_Y}\mathsf T_j(\rho_y,\alpha) +2\mathsf T_j(q_Y,\alpha). \end{align*}\] Its right side consists of explicit finite cut sums, so no selection of couplings across places or random grids is required. Integrating and using (36) and (38) gives \[ \left\langle \mathbb E_{x,x',y\ \mathrm{iid}\ \alpha} |D_j(x,y)-D_j(x',y)|\right\rangle=o(W). \tag{39}\]

A lower bound for tree distance fluctuations

We finish with an elementary fact whose constants do not depend on the shape of the tree.

In the tree for \(D_j\), every positive edge has both side masses greater than \(\delta/2\), by the support condition on \(w\). Apply Lemma 9 with \(L=\ell_j\), sum over \(j=1,2\), and average over places and grids. Equations (31) and (39) give \(W\ll o(W)\), a contradiction. This excludes every subsequence on which \(W\to\infty\).

A selected embedding and normalized limits

At an archimedean place \(v\), define the nonnegative local scale \[T_v=\mathbb E_{\rm grids}\sum_{j=1}^2 \bigl(\mathbb E_\pi L_j+\mathbb E_\alpha S_j\bigr).\] Let \(V_v\) be the sum of the local integrals on the left of (19), for the fiber family and the retained incoming family, including both coordinates and grid expectations but omitting the place weight. Since \(W\) is bounded, Lemma 8 gives \[\sum_{v\mid\infty}\sigma_vT_v\le C,\qquad \sum_{v\mid\infty}\sigma_vV_v=\eta_n\longrightarrow0.\] The archimedean weights sum to one. The places with \(T_v\le2C+1\) have total weight at least \(1/2\), so one of them satisfies \(V_v\le2\eta_n\). Choose such a place for each \(n\). This argument requires no lower bound on an individual place weight. Henceforth all coordinates and grids in this section refer to the selected embedding.

Equation (10) and the grid error bound (9) imply, for \(j=1,2\), \[\mathbb E_{\rm grids}\mathbb E_\pi|i_j(x,y)-M_j| \le \mathbb E_{\rm grids}\mathbb E_\pi \bigl(L_j(x,y)+S_j(x)+S_j(y)+|\Delta_j(x,y)|\bigr) =O(1).\] Set \(m_j=\mathbb E_\pi i_j(x,y)\). For each realized grid, \(|m_j-M_j|\le\mathbb E_\pi|i_j-M_j|\), and thus \(\mathbb E_\pi|i_j-m_j|=O(1)\). Normalize the coordinates by \[U_j(x)=e^{m_j}Z_j(x)+b_j,\] where the translation \(b_j\) will be chosen below. We have \[ \mathbb E_\pi\bigl|\log|U_j(x)-U_j(y)|\bigr|\le C_1. \tag{40}\] Only distinct pairs occur here.

Write \(\alpha_n^j=(U_j)_*\alpha\) for the empirical coordinate laws. Equation (40) implies \[(\alpha_n^j\otimes\alpha_n^j)\{|z-z'|>R\} \le \frac{C_1}{\log R}\qquad(R>1).\] Choose a fixed \(R_0>1\) making this bound less than \(1/2\). Averaging over a center at an empirical point gives a radius-\(R_0\) disk containing at least half the mass. Choose \(b_j\) to move that center to zero. If \(R>2R_0\), pairs between this disk and the complement of the radius-\(R\) disk have distance at least \(R/2\), so \[ \alpha_n^j\{|z|>R\}\le \frac{2C_1}{\log(R/2)}. \tag{41}\] For every center \(z_0\in\mathbb C\) and \(0<r<1/2\), a similar use of (40) gives \[ \alpha_n^j(B(z_0,r))^2 \le \frac1n+\frac{C_1}{\log(1/(2r))}. \tag{42}\] Indeed the diagonal mass of independent pairs in the ball is at most \(1/n\), and every remaining pair has distance at most \(2r\).

Let \(S=\widehat{\mathbb C}\) be the Riemann sphere with its chordal metric, and write \(\mathcal P(S)\) for its probability measures. The space of probability measures on a compact metric space is compact and metrizable for weak convergence (Gaans 2003, Proposition 5.3 and Theorem 4.2). We also use the open-set inequality in the Portmanteau theorem (Gaans 2003, Theorem 3.2). Passing to a common subsequence, let \(\alpha_n^j\Rightarrow\beta_j\) on \(S\), for \(j=1,2\). Apply Portmanteau to the open spherical sets \(\{|z|>R\}\cup\{\infty\}\) in (41); then let \(R\to\infty\). This shows \(\beta_j\{\infty\}=0\). Apply it to open disks in (42), then let \(r\downarrow0\). Each \(\beta_j\) is therefore a nonatomic probability measure on \(\mathbb C\).

The variance estimate determines the limiting laws

For either family from Lemma 8, put \(\nu_n^j=(U_j)_*\nu\), and write \(\mu\varphi=\int\varphi\,d\mu\) for a measure \(\mu\) and a continuous test \(\varphi\). We claim that every \(\varphi\in C(S)\) satisfies \[ \mathbb E_{\mathcal F} \left|\int\varphi\,d\nu_n^j-\int\varphi\,d\beta_j\right| \longrightarrow0. \tag{43}\] The family weights may sum to less than one; each member \(\nu\) is still a probability measure. In both applications, \(\mathbb E_{\mathcal F}\nu\le B\alpha\) for a fixed \(B\).

Here are the details converting integrated cell variance into (43). Push every realized grid through the same normalization \(z\mapsto e^{m_j}z+b_j\). This shifts its levels by \(-m_j\) and leaves integration over all levels unchanged. Fix \(R>0\) and \(0<\epsilon<1\). At normalized levels in the interval \[I_\epsilon=[-\log\epsilon,-\log\epsilon+1],\] the actual square cells have sides between \(\epsilon/(2e)\) and \(\epsilon\). At most \(N=N(R,\epsilon)\) squares meet the radius-\(R\) disk, uniformly in the grid and the level. For each such actual cell \(B'\), let \(C\) be its transformed occurrence: \(C=B'\) unless \(B'\) is the giant, in which case \(C=P\setminus B'\). Since both measures have mass one, \[|\nu(B')-\alpha(B')|=|\nu(C)-\alpha(C)|.\] Occurrences are retained with multiplicity. For those with \(|C|>n^{1-c}\), Cauchy–Schwarz over the family, grids, unit level interval, and at most \(N\) occurrences gives an integrated sum of mean absolute discrepancies at most \(\sqrt{NV_v}\). For the others, including an empty giant complement, \[\mathbb E_{\mathcal F}|\nu(C)-\alpha(C)| \le(B+1)\alpha(C)\le(B+1)n^{-c}.\]

Approximate \(\varphi\) on each of these squares by its value at any point of the square. Its oscillation is at most \(\omega_\varphi(\sqrt2\epsilon)\), where \(\omega_\varphi\) is a uniform modulus of continuity in the spherical metric. The remaining squares lie outside the radius-\(R\) disk, and their mean family mass is at most \(B\alpha_n^j\{|z|>R\}\). Integrating the resulting test-function estimate over grids and the unit interval gives \[\begin{align*} \mathbb E_{\mathcal F}|\nu_n^j\varphi-\alpha_n^j\varphi| \le{}& 2\omega_\varphi(\sqrt2\epsilon) +(B+1)\|\varphi\|_\infty\alpha_n^j\{|z|>R\}\\ &+\|\varphi\|_\infty \bigl(\sqrt{NV_v}+(B+1)Nn^{-c}\bigr). \tag{44}\end{align*}\] For fixed \(R,\epsilon\), let \(n\to\infty\); then let \(R\to\infty\) and \(\epsilon\downarrow0\). Equation (41) and \(\alpha_n^j\Rightarrow\beta_j\) prove (43). For the incoming laws, restoring all omitted targets changes a mean test discrepancy by at most \(2\|\varphi\|_\infty\) times their \(o(1)\) total \(q_Y\)-mass. Thus (43) holds in full \(q_Y\)-expectation for \(\rho_y\) as well.

We have obtained the same nonatomic coordinate limits for typical fibers and for typical incoming laws. The first conclusion will preserve the fiber equation in a limit; the second will allow its source coordinate to vary independently of its target.

The joint limit and its fiber equation

For a \(q\)-edge \((x,y)\) in fiber \(e\), consider the joint random object \[\bigl(U_1(x),U_2(x),U_1(y),U_2(y), (U_1)_*\mu_e,(U_2)_*\mu_e\bigr) \ \in\ S^4\times\mathcal P(S)^2.\] Take a subsequential weak limit \(Q\), and denote its four coordinate entries by \((a,b,a',b')\). The last two entries equal \((\beta_1,\beta_2)\) almost surely. To justify this simultaneously for all tests, choose a countable dense sequence \((f_r)\) in the unit ball of \(C(S)\). The metric \[d(\mu,\beta)=\sum_{r\ge1}2^{-r}|\mu f_r-\beta f_r|\] defines weak convergence. Equation (43) on finitely many tests, followed by a bound on the series tail, gives convergence of this metric to zero in mean for each fiber projection. Passing to \(Q\) proves the assertion.

The inequality \(q\le B_0\pi\le2B_0\alpha\otimes\alpha\) implies that each limiting same-coordinate pair law is dominated by \(2B_0\beta_j\otimes\beta_j\). Indeed the inequality passes to the limit against every nonnegative continuous test on \(S^2\). Since \(\beta_j\) is nonatomic and gives infinity mass zero, we have \[a,b,a',b'\in\mathbb C,\qquad a\ne a',\quad b\ne b', \qquad a'\in\mathop{\mathrm{supp}}\beta_1,\quad b'\in\mathop{\mathrm{supp}}\beta_2 \quad\text{$Q$-almost surely}.\]

Moreover, \(a\) has law \(\beta_1\) independently of the joint pair \((a',b')\). For \(f\in C(S)\) and \(g\in C(S^2)\), conditioning on the actual target vertex gives \[\begin{align*} &\left|\mathbb E_q f(U_1(x))g(U_1(y),U_2(y)) -(\beta_1f)\mathbb E_{q_Y}g(U_1(y),U_2(y))\right|\\ &\hspace{15mm}\le \|g\|_\infty\mathbb E_{q_Y} \left|\int f\,d(U_1)_*\rho_y-\beta_1f\right| \longrightarrow0 \end{align*}\] by (43). Such products determine the joint measure on \(S\times S^2\), proving this independence. Keeping both target coordinates together in \(g\) is what allows us to fix the entire target pair in the final obstruction.

On the open domain of finite quadruples with \(a\ne a'\) and \(b\ne b'\), put \(t'=(a'-a)(b'-b)\). We next obtain \[ \beta_2=\Phi_*\beta_1,\qquad \Phi(z)=b+\frac{t'}{z-a} \quad(z\in S), \qquad Q\text{-almost surely}. \tag{45}\] For each prelimit edge, its two fiber projection measures are related by this Möbius formula, with its four coordinates in place of \((a,b,a',b')\). Indeed, normalization multiplies the original fiber product by \(e^{m_1+m_2}\), and evaluation at \(y\) identifies the new product as \((U_1(y)-U_1(x))(U_2(y)-U_2(x))\).

To pass the equality to the limit, note that the matrix \[\begin{pmatrix}b&t'-ab\\1&-a\end{pmatrix}\] represents \(\Phi\) on the sphere and has determinant \(-t'\ne0\). The map is jointly continuous in its parameters and sphere variable, including at its pole. Thus its pushforward acts continuously on probability measures with the weak topology, as is seen by testing continuous functions and using uniform continuity on compact parameter neighborhoods. The equality graph is relatively closed over the stated open domain. Its complement within that domain is open in the ambient joint space and has zero mass in every prelimit law. Portmanteau gives it zero \(Q\)-mass. The domain itself has full \(Q\)-mass, proving (45).

A nonatomic obstruction to the fiber maps

The following elementary observation completes the argument. The second source coordinate \(b\) may depend on the other coordinates. In the proof it contributes only an additive constant to a logarithmic distance, so the independence already proved for \(a\) is sufficient.

The law \(Q\) constructed above satisfies every hypothesis of Lemma 10, so a bounded-\(W\) subsequence is impossible. Section 6 ruled out a subsequence with \(W\to\infty\). Every sequence of nonnegative real numbers has a bounded subsequence or a subsequence tending to infinity. Therefore the graph sequence from Proposition 4 cannot exist. The assumption that \(\Theta(s)>0\) somewhere was false, so \(F_n(s)\to0\) for each fixed \(s>0\). This proves Theorem 1.