Every finite point set P in the real plane carries a collection of
distinct distances at each of its pins x in P. Two questions attach to
this data. How few distinct distances can a single pin see, and how
many pins can be bad at once. The second question is the sharper one,
and the answer below is that bad pins can never occupy a positive
fraction of a large configuration.
For x in P, write
D_x(P) = { |y - x| : y in P, y != x }.
Erdos asked in 1957 whether some pin always sees at least n^(1 - eps)
distinct distances, for every fixed eps > 0. The question is Problem
16 of Some unsolved problems (Michigan Math J 4(3):291-300, 1957).
./erdos1957-unsolved-prob.pdf
The theorem below proves a strictly stronger statement: at all but
o(n) pins the count reaches n^(1 - eps).
The right object to track is not just the count but the multiplicity
of any single distance at a pin. For distinct x, y in P define
k_P(x,y) = #{ z in P \ {x} : |z - x| = |y - x| }.
So k_P(x,y) counts every point at the same distance from x as y is,
including y itself. Different pairs use their own source point, and
two pairs with the same numerical distance but different sources are
treated separately.
For n >= 2 and s > 0 put
F_n(s) = sup_{ P subset R^2, |P| = n }
( #{ (x,y) in P^2 : x != y, k_P(x,y) >= n^s }
/ ( n (n - 1) ) ).
Theorem. For every fixed s > 0, F_n(s) -> 0 as n -> infinity.
The convergence is uniform: no separation, general position, or
coordinate hypothesis is imposed, and the constants only depend on s.
The exponent s is fixed; the theorem supplies no explicit rate of
decay.
Corollary. For every fixed eps > 0,
sup_{ P subset R^2, |P| = n }
( #{ x in P : |D_x(P)| < n^(1 - eps) } / n ) -> 0.
In particular every sufficiently large n-point planar set has a pin
determining at least n^(1 - eps) distinct nonzero distances.
Proof of the corollary. For eps >= 1 the assertion is trivial; every
pin with n >= 2 sees at least one distance. Suppose 0 < eps < 1 and
take s = eps/2. A pin with fewer than n^(1 - eps) distance classes
can place at most n^(1 - eps + s) neighbours in classes of size below
n^s, so it contributes at least n - 1 - n^(1 - eps/2) pairs counted
by F_n(s). If there are b such bad pins, then for all large n
b/n <= (n-1) F_n(eps/2) / ( n - 1 - n^(1 - eps/2) ) -> 0.
The bound is uniform in P.
The corollary is nearly sharp. If P consists of the vertices of a
regular polygon together with its centre, the centre sees a single
nonzero distance. Exceptional pins are unavoidable; the theorem only
says that such pins cannot fill a positive fraction of the
population.
History and external context. Erdos opened both the distinct-distance
and the repeated-distance questions in 1946.
./erdos1946-amm-distances-n.pdf
The original note introduces the quantity that later became the
object of study. Denote by f(n) the least number such that any set of
n points of the plane contains some point that determines at least
f(n) distinct distances to the remaining points. The note points out
that the integer square lattice provides sets for which the union of
all such distances has size O(n / sqrt(log n)). The present question
fixes one source and measures the count at that source. In that
reformulation the lattice is not the extremal example; a symmetric
configuration close to a regular polygon can make a single pin see
just one distance.
Guth and Katz proved that every planar n-point set determines at
least a constant times n / log n distinct distances.
./guth-katz-2015-annals.pdf
Their conclusion concerns the union over all sources; the pinned
version fixes the source and asks for the worst single-source count
in the same configuration.
Pinned bounds grew historically by combining incidence geometry with
progressively sharper counting. Solymosi and Toth obtained the order
n^(6/7).
./solymosi-toth-2001.pdf
Tardos improved the exponent with an entropy inequality.
./tardos2003-distinct-sums.pdf
Katz and Tardos pushed every exponent below
(48 - 14 e) / (55 - 16 e) = 0.864137...
in their note A new entropy inequality for the Erdos distance
problem.
./katz-tardos-2004.pdf
Each of these papers guarantees that some pin in every configuration
is good. The sharper pinned conjecture asks for order n / sqrt(log n)
distances at a pin; the introduction of Bisectors and pinned distances
documents the distinction between the real planar problem and
pinned-distance theorems over other fields.
./lund-petridis-2020.pdf
The almost-every-pin corollary above resolves the weak pinned
conjecture in the form in which it appears as Conjecture 4.7 of
Generalised Erdos distance theory on graphs.
./dewar-etal-2025-weak-pinned.pdf
Outline of the argument. Assume the theorem fails. Then a positive
density of ordered distinct pairs have sources whose distance fiber is
as large as n^s. Record these pairs as a directed graph, and let theta
be the limiting edge density. A deletion procedure on the graph forces
both a density bound from below and a fiber size bound from above,
giving good control of large and small subsets.
Work inside a number field K subset C that contains the coordinates of
every point. Write x = (u,v) and set Z_1(x) = u + iv and Z_2(x) = u -
iv, with i = sqrt(-1). The squared distance factors exactly:
|x - y|^2 = ( Z_1(y) - Z_1(x) ) ( Z_2(y) - Z_2(x) ).
The product formula in K, applied to both coordinates, balances the
logarithms of these differences across all places, so on a single
fiber the sum of the two coordinate depths is a fixed constant.
To turn logarithms of distances into integers, nested partitions are
built from randomly shifted square grids at each complex place. The
grid depth differs from the raw logarithmic depth by a small error
Delta whose distribution is independent of the pair and which has an
exponential tail. Replacing any cell with more than half the points by
its complement keeps the cell size balanced and produces an exact
additive identity: the total overlap of two distinct points, summed
over all levels, all coordinates and all places, equals a function of
the first point plus a function of the second point plus a constant.
That identity produces a nonnegative scale W, the expected overlap of
two distinct uniform points plus a correction for giant complements.
Two facts pin down W. The variance estimate compares the fiber law and
the uniform law cell by cell and shows that cells of size at most
n^(1-c) carry only o(W+1) of the overlap; the same variance estimate
compares the incoming-neighbour law at each target with the uniform
law.
Then W either stays bounded or grows without bound along some
subsequence. In the unbounded case, cells of mass bounded away from
zero define weighted trees; transporting mass along these trees uses a
classical distance formula for trees, and comparison of the fiber law
with the uniform law turns the fiber equation into an impossibility
for three independent uniform vertices. In the bounded case, one
complex embedding is normalized so that the two coordinate laws have
nonatomic weak limits; the variance estimate preserves the fiber
equation in the limit and yields a family of Mobius maps between the
two limits. A short order-statistics argument rules out any such
family when both limit laws are nonatomic. Both alternatives are
impossible, hence the theorem.
The proof of the main theorem begins. Assume for the rest of the
argument that the conclusion fails at some exponent: for at least one
positive t the numbers F_n(t) do not tend to zero. Define
Theta(t) = limsup_{n -> infinity} F_n(t),
s_* = sup { t > 0 : Theta(t) > 0 }.
Each F_n, and hence Theta, is nonincreasing. And F_n(t) = 0 vanishes
for t >= 1 because a distance class at a pin has at most n - 1
points. So 0 < s_* <= 1. Assuming the theorem fails means Theta(t) >
0 for some positive t, hence s_* > 0. Fix
0 < c < s_*/3,
max { c, (1-c) s_* } < s < s_*,
c < gamma < s,
choosing s to be a continuity point of Theta. Such an s exists
because a monotone function has at most countably many
discontinuities. Set theta = Theta(s) > 0; positivity follows from
s < s_* and monotonicity of Theta. All asymptotic estimates below are
with respect to the sequence of configurations constructed in the
next step; their constants may depend on (c, s, gamma, theta) but not
on n or on the configurations.
Two types of probability laws at a graph. Once the directed graph E
of large-fiber pairs is chosen, several probability measures attached
to it will appear repeatedly. The piece of terminology that makes
them usable is fiber: two edges lie in the same fiber when they share
their source and their squared distance. Write x_e for the source of
a fiber e, F_e subset of P minus {x_e} for its target set, and k_e =
|F_e|. Let d^-(y) denote the number of incoming edges at y.
The laws are these.
alpha: uniform probability measure on P.
pi: uniform on the n(n-1) distinct ordered pairs in P.
q: uniform on E, with source and target marginals q_X and q_Y.
mu_e: uniform on F_e.
rho_y: uniform on the incoming neighbours of y, defined when
d^-(y) > 0.
To sample q, choose a fiber e with probability k_e / |E| and then
choose y with law mu_e, setting x = x_e. Equivalently, sample y with
law q_Y and then x with law rho_y. These laws are attached to the
finite graph; they do not depend on any of the absolute values
introduced later.
The extraction of the graph from the failure of the theorem controls
not only its density and fiber sizes but also how much of a fiber can
sit inside a small subset of vertices. That last control is what lets
fibers be compared with the uniform distribution.
Tail lemma. If r_N -> s and M_N -> infinity, then
sup_{ m >= M_N } ( F_m(r_N) - theta )_+ -> 0,
where the supremum is over integers m >= 2 and (u)_+ = max { u, 0 }.
Proof. Fix eps > 0. Continuity of Theta at s gives delta > 0 with
Theta(s - delta) < theta + eps/2. By the definition of limsup, F_m(s
- delta) <= Theta(s - delta) + eps/2 for all large m. For all large
N, r_N >= s - delta, and every m >= M_N then lands in this range.
Monotonicity of F_m in its argument gives F_m(r_N) <= theta + eps.
Extremal graph proposition. Under the present supposition there is a
sequence of finite sets P subset of R^2 of real algebraic points,
with n = |P| -> infinity, and nonempty directed graphs E on P, with
these properties. Their densities d_n = |E| / [n (n-1)] tend to
theta, and every fiber has k_e >= n^(s - o(1)), uniformly in e. There
is a fixed constant B_0 such that, pointwise,
q <= B_0 pi, q_X, q_Y <= B_0 alpha,
(q_X + q_Y)/2 >= (1 - o(1)) alpha,
and
q_Y { y : d^-(y) < n^gamma } = O(n^(gamma - 1)) = o(1).
Put lambda_n = 1/log n. For every subset C subset of P, writing m =
|C| and p_e = mu_e(C),
E_e [ 1_C(x_e) p_e 1_{ p_e >= lambda_n } ]
<= (1 + o(1)) pi(C^2) when m > n^(1-c),
<= o(1) pi(C^2) when m <= n^(1-c).
All errors are uniform in vertices, fibers and subsets.
Proof of the proposition. Three steps: passing to a sequence of
algebraic configurations that preserves every distance equality;
deleting low-degree vertices and small fibers; and the subset
estimate.
Step 1. For fixed N, the numerator defining F_N(s) takes values in
the finite set {0, 1, ..., N(N-1)}, so it attains its maximum on some
configuration. Choose N -> infinity with F_N(s) -> theta and one
attaining configuration for each N.
That configuration can be replaced by one with real algebraic
coordinates without changing any distance class. Label its points
1, ..., N and write the coordinate variables as (u_i, v_i), with
D_{ij} = (u_i - u_j)^2 + (v_i - v_j)^2. Require D_{ij} > 0 for i <
j, and prescribe for every pair of unordered pairs either the
equality D_{ij} = D_{kl} or the disequality D_{ij} != D_{kl} that
holds in the original configuration. These are finitely many
polynomial conditions over Q: a disequality can be encoded as
(D_{ij} - D_{kl})^2 > 0. Their existential sentence holds over R, so
by transfer for real closed fields it holds over the real algebraic
numbers. The standard reference for real closed fields and their
model theory is Lecture 9 of Kuhlmann's notes.
./kuhlmann-2009-rcf-notes09.pdf
The resulting points are distinct and retain the exact
distance-equality pattern, so every fiber size is unchanged and the
value of F_N(s) is preserved. The real threshold N^s need not enter
the transferred system: it only compares against the integer class
sizes, which are preserved. No bound on the degrees or the heights of
the algebraic coordinates is imposed.
Step 2. On this algebraic configuration, keep every edge counted by
F_N(s). Each initial fiber has at least N^s edges. Put
K_N = N^s / log N,
f_N = log K_N / log N = s - (log log N) / log N,
m_0 = ceil(N / 2).
For all large N, f_N > 0. The tail lemma gives
b_N := max_{ m_0 <= m <= N } ( F_m(f_N) - theta )_+ -> 0.
Set
t_N = b_N + |F_N(s) - theta| + 1 / log N,
eta_N = sqrt(t_N),
so eta_N -> 0 and eventually theta - eta_N > 0.
Whenever a remaining nonempty fiber has fewer than K_N edges, discard
it. If the current vertex count is m and a vertex has combined in-
and out-degree below 2 (theta - eta_N) (m - 1), delete that vertex
and all its incident edges. Clean out small fibers after every vertex
deletion and continue until all degree lower bounds hold.
There are at most N(N-1)/N^s initial fibers. Restricting the vertex
set never splits a source-distance fiber, and each initial fiber can
be discarded at most once, costing fewer than K_N edges each. All
fiber cleanups together therefore cost at most N(N-1)/log N edges.
If the deletion process reaches m_0 vertices, write A_N = N(N-1) and
B_N = m_0(m_0 - 1). The vertex deletions cost fewer than
2 (theta - eta_N) sum_{ m = m_0+1 }^N (m - 1)
= (theta - eta_N) (A_N - B_N)
edges. After all cleanups, at least
F_N(s) A_N - (theta - eta_N) (A_N - B_N) - A_N / log N
edges remain. Their excess over (theta + b_N) B_N is at least
eta_N (A_N - B_N) - t_N A_N
= eta_N A_N ( 1 - B_N/A_N - eta_N ) > 0
for large N, because B_N/A_N -> 1/4. But every remaining edge has a
fiber of at least K_N = N^{f_N} >= m_0^{f_N} edges, whose density is
at most F_{m_0}(f_N) <= theta + b_N, a contradiction.
So the process stops on a set P of size n > m_0 with a graph E whose
fibers all have size at least K_N. Summing the degree lower bounds and
using the definition of b_N gives
theta - eta_N <= d_n <= F_n(f_N) <= theta + b_N.
The upper bound applies because K_N >= n^{f_N}. Thus d_n -> theta,
and k_e >= K_N >= n^{s - o(1)} uniformly in e.
Step 3. For every edge, q(x,y)/pi(x,y) = 1/d_n, so B_0 = 2/theta
works in the first inequality for large n. Summing over one
coordinate gives the marginal bounds. At every vertex the degree lower
bound gives
(q_X(x) + q_Y(x))/2
>= ((theta - eta_N)/d_n) alpha(x)
>= ((theta - eta_N)/(theta + b_N)) alpha(x)
= (1 - o(1)) alpha(x).
Only the sum of the two marginals is bounded below; separate lower
bounds are not needed. For targets of small incoming degree,
q_Y { y : d^-(y) < n^gamma }
<= n n^gamma / |E| = O(n^(gamma - 1)),
which is o(1) since gamma < s < 1.
Step 4. Define
T_n = lambda_n K_N = N^s / (log N log n).
Since N/2 < n <= N, there is a sequence delta_n -> 0 with T_n >=
n^{s - delta_n}; in particular T_n -> infinity. Multiplying the left
side of the subset estimate by |E| counts exactly the edges (x_e, y)
in C x C with |F_e cap C| >= lambda_n k_e. Every such pair satisfies
k_C(x_e, y) >= |F_e cap C| >= T_n.
A nonzero contribution therefore needs m - 1 >= T_n, because the
source sits in C and is excluded from its own distance class. Hence
all bounded subsets contribute zero for large n; the cases m = 0 and
m = 1 give zero at every n.
For m > n^{1-c}, the positive exponent s - delta_n gives T_n >=
n^{s - delta_n} >= m^{s - delta_n}. The tail lemma gives F_m(s -
delta_n) <= theta + o(1) uniformly in these sizes. Divide the pair
count by |E|:
E_e [ 1_C(x_e) p_e 1_{ p_e >= lambda_n } ]
<= ((theta + o(1))/d_n) (m(m-1)/(n(n-1)))
= (1 + o(1)) pi(C^2).
For m <= n^{1-c}, choose a fixed s' with s_* < s' < s/(1-c),
possible because s < s_* and c < s_*/3. For large n,
T_n >= n^{s - delta_n} >= n^{(1-c) s'} >= m^{s'}.
The definition of s_* gives Theta(s') = 0, so F_m(s') -> 0 as m ->
infinity. Every contributing subset has m >= T_n + 1, so
sup_{ m >= T_n + 1 } F_m(s') -> 0,
again only integer sizes considered. Dividing this pair count by |E|
gives o(1) pi(C^2) uniformly over all contributing small subsets; the
other subsets contribute zero. This also covers s' > 1, when the
corresponding pair count is identically zero.
From now on the sequence P and its graph E from the proposition are
fixed. In particular every k_e >= n^gamma for large n. The graph
and its sampling laws are now fixed.
The arithmetic model. The remaining proof is arithmetic. The two
coordinate maps and the nested grids are built here; they turn
logarithms of distances into sums of nonnegative overlaps.
Two coordinate maps. Take one algebraic configuration from the
extremal-graph proposition. Fix a number field K subset of C
containing all its coordinates and the imaginary unit i = sqrt(-1).
For a point x = (u,v) put
Z_1(x) = u + iv, Z_2(x) = u - iv.
Both maps are injective, because u and v are real in the chosen
embedding. At another complex embedding K -> C the two coordinates
need not be complex conjugates; their injectivity and the algebraic
factorisation are the only properties used. For distinct x, y the
squared distance factors exactly in K:
t(x,y) = ( Z_1(y) - Z_1(x) ) ( Z_2(y) - Z_2(x) ).
Place-by-place depth. 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 let sigma_v = [K_v : Q_p] / [K : Q]. At an archimedean
place v, all of which are complex since i is in K, use complex
modulus and sigma_v = 2/[K : Q]. The normalised product formula
(Milne, Algebraic Number Theory, Theorems 7.14-7.15 and Lemma 8.6)
reads
sum_v sigma_v log |z|_v = 0 (z in K, z != 0),
sum_{ v | infinity } sigma_v = 1.
./milne-ant-2020.pdf
The weights convert the norm-based finite absolute values and
squared complex moduli in that reference to the values used here,
after division by [K : Q]. Here and below logarithms are natural.
For a fixed configuration only finitely many finite places make any
coordinate difference a nonunit, because each nonzero element of K
has nonzero valuation at only finitely many prime ideals.
At a place v, define the raw depths
i_j(x,y) = -log |Z_j(x) - Z_j(y)|_v, x != y, j = 1, 2.
The place is suppressed in the notation. On a fiber F_e with source
x_e and squared distance t_e, the factorisation yields
i_1(x_e, y) + i_2(x_e, y) = h_e, h_e = -log |t_e|_v,
y in F_e.
The number h_e does not depend on the choice of y in the fiber. This
is the fiber equation; it is the entire reason the proof works.
Nested grids at complex places. At a finite place put d_j(x,y) =
i_j(x,y). The ultrametric inequality makes d_j(x,y) >= r an
equivalence relation. At a complex place the same nesting is
obtained from random square grids, with a controlled error.
Independently of all vertex and fiber sampling, choose a uniformly
random orientation and xi uniform on [0, log 2). Grid side lengths
are l_m = e^xi 2^m for m in Z, with l_0 = e^xi in [1,2). 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 >= 0, add 0 or l_m with
probability 1/2 each, 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.
This recursive shift construction appears in Arora's polynomial-time
approximation scheme for Euclidean TSP, Section 2.1, in the form of
randomly shifted quadtree grids.
./arora1998-euclidean-tsp.pdf
For distinct points let d_j(x,y) denote minus the logarithm of the
smallest grid side at which the two images share a cell. Put
d_j(x,x) = +infinity and, for x != y,
Delta_j(x,y) = d_j(x,y) - i_j(x,y).
The three depth quantities play distinct roles.
i_j(x,y): raw logarithmic depth, -log |Z_j(x) - Z_j(y)|_v.
d_j(x,y): sharing depth in the nested hierarchy.
Delta_j(x,y): hierarchy error, d_j(x,y) - i_j(x,y).
Grid error lemma. At every complex place the depths are almost surely
finite on distinct pairs. The distribution of Delta_j(x,y) is
independent of the distinct pair, of the place and of the
configuration. Its finite mean is an absolute constant kappa, and
absolute constants C_0, b_0 > 0 satisfy
Pr( |Delta_j(x,y)| > u ) <= C_0 e^{-b_0 u} (u >= 0).
At finite places Delta_j = 0.
Proof. Write R > 0 for the separation of the two complex values and
(a,b) for their displacement along the oriented axes. At side length
l a uniform shift gives a sharing probability of
(1 - |a|/l)_+ (1 - |b|/l)_+.
Sharing is impossible if l < R/sqrt(2), while separation has
probability at most (|a| + |b|)/l <= sqrt(2) R/l. The separation
events shrink as the grids coarsen, with probabilities going to zero,
so almost surely the pair eventually shares a cell and there is a
smallest sharing scale.
Write Delta = d_j(x,y) + log R. A sharing side gives Delta <=
log sqrt(2). For u >= 0 let l be the largest grid side no larger than
R e^u, so l > R e^u / 2. On {Delta < -u} the pair is separated at
that side, so
Pr(Delta < -u) <= 2 sqrt(2) e^{-u}.
This proves the tail and a uniform first absolute moment.
For any real u the event {Delta >= u} is sharing at the largest side
l <= R e^{-u}. The law of l/R depends only on u, by the uniform
logarithmic phase, and the displacement direction relative to the
axes is uniform. The sharing formula therefore makes the entire law
of Delta independent of the pair. Only countably many random
variables and cell tests are used, so all these events are
measurable.
Removing the giant cell. Discard the null event on which any
distinct-pair depth is not finite. For a place and coordinate j let
Pi_j(r) denote the partition with sharing criterion d_j(x,y) >= r,
for r in R. These partitions are nested as r grows. Call a cell
giant if its size exceeds n/2, and denote such a cell, when it
exists, by G_j(r). Replace the giant by its complement and keep all
other cells. The resulting C_j(r) is a multiset: if the complement
already occurs as another cell, it occurs twice. Every C in C_j(r)
has alpha(C) <= 1/2. Multiplicities matter because the entire
argument is an integral of pairwise overlaps and no term can be
discarded.
This replacement removes the overlap of all pairs at very coarse
scales. The cost is a function of each endpoint separately, which
cancels when pair laws with the same combined marginals are
compared. On a fixed realised hierarchy define
M_j = sup { r : G_j(r) exists },
S_j(x) = int over { r : G_j(r) exists } 1_{ x not in G_j(r) } dr,
L_j(x,y) = int_R sum_{ C in C_j(r) } 1_C(x) 1_C(y) dr,
x != y.
Additive identity for a fixed hierarchy. For each realised hierarchy
on n >= 2 points the displayed quantities are finite, and for
distinct x, y,
d_j(x,y) = M_j + L_j(x,y) - S_j(x) - S_j(y).
They are integrable over the grids. At every finite place where all
distinct coordinate differences are units, M_j = S_j = L_j = 0.
Proof. Suppress j. Let d_- and d_+ be the minimum and maximum of the
finitely many distinct-pair depths. Below d_- the partition has only
the cell P; above d_+ it has only singletons. A giant at any level
has a giant ancestor at every coarser level, so a giant exists on an
initial ray with endpoint M in [d_-, d_+], up to an immaterial
endpoint convention.
Let o(r) = 1_{ d(x,y) >= r }, g(r) = 1_{ G(r) exists }, and
t(r) = sum_{ C in C(r) } 1_C(x) 1_C(y). Expanding the giant
indicator through its complement gives, pointwise,
o(r) - g(r) = t(r)
- 1_{ G(r) exists, x not in G(r) }
- 1_{ G(r) exists, y not in G(r) }.
The integral of the left side is d(x,y) - M. Each term on the right
vanishes outside [d_-, d_+] on a distinct pair, and t(r) <= 2.
Integrating proves the identity, with bounds
0 <= S(x) <= d_+ - d_-,
0 <= L(x,y) <= 2 (d_+ - d_-),
|M| <= max { |d_-|, |d_+| }.
These are grid-integrable by the grid error lemma, since the
configuration has finitely many pairs and finite raw depths. At a
finite place with all raw depths zero, d_- = d_+ = 0, proving the
final assertion. Ties affect neither the identity nor any integral.
The empty complement of P and singleton occurrences contribute zero
to every distinct-pair overlap. L_j(x,x) is not defined: singleton
cells persist to arbitrarily fine scales, so that integral would
diverge. This is the only place where the off-diagonal restriction
matters, and it is why every subsequent estimate keeps a strict
distinct-pair convention.
Averaging over places and grids. For a nonnegative or integrable
quantity depending on the place and grids, write
avg(H) = sum_v sigma_v E_grids H,
I(g) = avg( sum_{ j=1 }^2 int_R sum_{ C in C_j(r) } g dr ).
There is no grid randomness at finite places. The integrand g may
depend on the coordinate, level and cell occurrence. All sums over
C_j(r) retain multiplicities. Set
S^*(x) = avg( S_1(x) + S_2(x) ),
L^*(x,y) = avg( L_1(x,y) + L_2(x,y) ), x != y.
The additive-identity lemma and the finite set of nonunit places show
that these and avg( M_1 + M_2 ) are finite. For each distinct pair the
product formula and the grid error lemma give
avg( d_1(x,y) + d_2(x,y) ) = 2 kappa.
Taking expectations in the depth identity is therefore legitimate
and gives an additive identity,
L^*(x,y) = b_* + S^*(x) + S^*(y), x != y,
with b_* = 2 kappa - avg( M_1 + M_2 ). In particular a signed law on
distinct pairs with zero total mass and zero sum of its two
marginals integrates to zero against L^*. This consequence is exact;
no diagonal sampling error is involved.
The three nonnegative finite scales used below are
W_2 = I( pi(C^2) ) = E_pi L^*(x,y),
A = E_alpha S^*(x),
W = W_2 + A.
The quantity W_2 measures distinct-pair overlap after the giant is
replaced; A measures the average time spent outside the giant. The