Number fields and rare planar distances

../

evidence/: Exact distance spectra for all nonempty subsets of a four-by-four grid after two algebraic maps, and a seven-point example with a rich distance.

This lead's notes that cite no source are author-recorded research, not claims.

Target and standing

For a finite set $A\subset\mathbb R^2$, let $n=|A|$, let $m_A(r)$ count unordered pairs at distance $r>0$, and put

The intended questions of Problem 132 are whether $R(A)\ge2$ for every $n\ge5$, and whether the minimum of $R(A)$ over all $n$-point sets tends to infinity. The restriction $n\ge5$ is needed for the first question: two unit equilateral triangles sharing an edge give a four-point set with five unit pairs and one diametral pair.

This is an author-recorded research sketch, without independent mathematical review or a novelty claim. The number-field reduction does not solve either question. The initial approach separates coefficients of squared distances; the other approach transports whole distance classes through real embeddings. The outstanding issue is a bound that holds without choosing a favorable coordinate pattern or field embedding.

Proof sketch

1. Encode the equalities and inequalities between squared pair distances as a finite polynomial system over $\mathbb Q$. Real-closed-field transfer realizes the same pattern over the real algebraic numbers. Its finitely many coordinates generate a number field. This preserves multiplicities, but supplies no useful field-degree or coefficient-independence bound.
2. For a real number field $F$ and a real algebraic $s$ with $s^2\notin F$, consider finite subsets of $F\times sF$. Equality of squared distances separates the horizontal and vertical squared differences. A geometric count of long horizontal displacements produces linearly many rare classes even though some other classes can exceed $n$.
3. For coordinates in a number field, each real embedding gives a planar realization with exactly the same multiplicity classes. A class that is a diameter in any such realization has multiplicity at most $n$. Distinct conjugate diameters would therefore give distinct rare classes.
4. Neither coordinate separation nor distinct conjugate diameters is known here for arbitrary configurations. Generic deformation can split distance classes, so its rare classes cannot simply be transported back to the original metric.

The coordinate-separation argument below is complete as an author-recorded argument. The last step to arbitrary configurations remains open. The problem assessment records the literature search; this note claims no improvement to the unrestricted problem's standing.

A linear bound when distances separate the coordinates

Suppose a finite planar set $A$ has the following property: whenever two pairs have the same Euclidean distance, their absolute horizontal displacements agree and their absolute vertical displacements agree. Then

This condition permits repeated distances in two reflected directions; it does not require that every distance already have multiplicity at most $n$.

Here is a proof. The cases $n\le1$ are immediate. If all points lie on one vertical line, every distance graph is a disjoint union of paths, so each multiplicity is at most $n-1$. The distances from the lowest point supply at least $n-1$ distinct values. Assume otherwise that the horizontal range is $[a,b]$, with $W=b-a>0$. Choose points $p_L,p_R$ on the leftmost and rightmost vertical lines, and let $t$ be the number of points on the vertical midline $x=(a+b)/2$.

First consider any occurring distance with absolute horizontal displacement $u>W/2$. Every edge of its distance graph has horizontal displacement $u$, by the separation assumption. A vertex cannot have neighbors on both its left and its right: those neighbors would be horizontally $2u>W$ apart. On its one possible side, it has at most two neighbors, since the absolute vertical displacement is fixed. The graph has maximum degree at most two, and hence at most $n$ edges. This distance is rare in the required sense.

For every point right of the midline, take its distance to $p_L$; for every point left of it, take its distance to $p_R$. These $n-t$ assignments all have horizontal displacement greater than $W/2$. A particular resulting distance can be assigned by at most two points at each anchor, hence by at most four points altogether. This supplies at least

rare distances. If $t\ge1$, the $t$ midline points supply at least $t-1$ further distances, using distances from the lowest one. Their entire distance graphs are vertical by coordinate separation and are unions of paths. They therefore have multiplicity at most $n-1$. They are distinct from the preceding distances, whose horizontal displacement is positive. Thus, including the case $t=0$,

For $t=0$ this is at least $\lceil n/4\rceil$. For $t\ge1$, four times the right-hand side is at least $n-t+4(t-1)=n+3t-4\ge n-1$. This proves (1). The proof uses neither a grid shape nor a bound on the coordinate size.

Number-field realization

Let $F\subset\mathbb R$ be a number field, and let $s$ be real algebraic with $s^2\notin F$. Every finite subset of

satisfies the separation condition. Indeed, equality of two squared distances has the form

Linear independence of $1,s^2$ over $F$ gives $u^2=u'^2$ and $v^2=v'^2$, which is exactly the required equality of absolute coordinate displacements. Consequently (1) holds uniformly in $F$, $s$, and the finite subset. Taking $F=\mathbb Q$ and $s=1+\sqrt2$ gives the concrete family discussed in the question. For any number field $F$, a positive rational prime $p$ with $\sqrt p\notin F$ gives another choice $s=1+\sqrt p$; only finitely many quadratic fields can be subfields of $F$.

This proves the growth assertion on this restricted family. Bound (1) also gives two rare distances for $n\ge6$; it alone gives only one for $n=5$. This is not a solution for arbitrary planar sets.

The condition does not force all distances to be rare. For $s=1+\sqrt2$, take the images of the seven integer points

under $(x,y)\mapsto(x,sy)$. Their squared-distance multiplicities, written as pairs $(c,d)$ for $c+d\sqrt2$, are

Thus one distance occurs more than $n=7$ times, while five are rare.

Stronger separation by an algebraic shear

There is a simpler construction if all distance classes may be changed. Let $F\subset\mathbb R$ be a number field. Choose positive rational $a,b$ such that $[F(\sqrt a,\sqrt b):F]=4$, and map a finite $S\subset F^2$ by

Such choices exist: take $\sqrt a\notin F$ and then $\sqrt b\notin F(\sqrt a)$, using the finiteness of the quadratic subfields of each finite extension. The map is invertible over $\mathbb R$. The squared length of an image displacement is

The elements $1,\sqrt a,\sqrt b$ are linearly independent over $F$. Consequently $q(x,y)=q(x',y')$ forces equality of $y^2$, $xy$, and $x^2$. If $y\ne0$, this gives $(x',y')=\pm(x,y)$ with a common sign; if $y=0$, the same conclusion follows from $y'=0$ and $x'^2=x^2$.

Each distance graph on $T(S)$ therefore consists of edges with one fixed displacement up to sign. Its components are finite paths, so every multiplicity is at most $n-1$. Choose a linear functional taking distinct values on $S$, and a point where it is minimal. The $n-1$ displacements from that point cannot be equal or opposite, so their image lengths are distinct. Hence $R(T(S))\ge n-1$.

For $F=\mathbb Q$ one may use $a=2,b=3$. This is an explicit construction over $\mathbb Q(\sqrt2,\sqrt3)$, with no increasing-degree tower.

Why this deformation does not prove the original statement

The map can split an original distance class into many smaller classes. The inverse map can merge all those rare classes again. Real-closed-field transfer preserves the pattern; a generic algebraic shear does not.

Even preserving all distance equalities under an affine map can prohibit the required freedom. If the original set contains a unit square and the linear part of the map is $T$, put $M=T^{\mathsf T}T$. Equality of its horizontal and vertical sides forces $M_{11}=M_{22}$. Equality of its two diagonals forces $M_{12}=0$. Thus $M=cI$: every invertible affine map preserving these equalities is a similarity and leaves the multiplicity pattern unchanged. This obstruction already occurs in rational configurations of arbitrarily large size containing a square.

There is a useful limited converse. Suppose a finite configuration admits an invertible affine map preserving every equality of pair distances and $T^{\mathsf T}T$ is not a scalar matrix. Diagonalize that real symmetric matrix in orthonormal coordinates, with distinct eigenvalues $\lambda,\mu$. For equal-length displacements $(u,v),(u',v')$ we have both

Solving gives $u^2=u'^2$, $v^2=v'^2$. The original configuration therefore satisfies coordinate separation in these rotated axes, and (1) applies to it. Any counterexample to (1) must consequently have only similarities among its invertible affine maps preserving all distance equalities. This is an exact limit of this deformation route, not an obstruction to every arithmetic approach.

Conjugate diameters and a uniform obstruction

Let $A\subset F^2$ for a number field $F\subset\mathbb R$. For each real embedding $\sigma:F\hookrightarrow\mathbb R$, apply $\sigma$ to both coordinates. The resulting $A^\sigma$ has $n$ distinct points. A squared distance $q\in F$ becomes $\sigma(q)$, and injectivity of $\sigma$ preserves all equalities and inequalities of values in the sense of equality versus nonequality. Numerical order need not be preserved. Therefore corresponding distance classes have identical multiplicities.

Let $q_\sigma$ be the original squared-distance class that becomes the diameter of $A^\sigma$. The planar diameter bound gives

The same transport permits applying a known planar sufficient condition in any real embedding, such as the first-two-convex-layers criterion in Clemen--Dumitrescu--Liu, Theorem 1.3. This use is conditional on that external source theorem; no independent proof review of it is supplied here.

Increasing the field degree does not make the right-hand side of (2) grow. For any totally real number field $F$, choose a totally positive primitive element $\theta$: a sufficiently large rational translate of a primitive element works. Its distinct conjugates are all positive, so their squares are distinct and $\mathbb Q(\theta^2)=F$. Given $n\ge3$, choose an integer

and take

In every real embedding, the vertical coordinates lie strictly between zero and $1/2$. Distances between two vertical points are less than $1/2$; distances from an endpoint to a vertical point are less than $\sqrt{5/4}$. Thus the two horizontal endpoints are the unique diameter in every embedding, and the right-hand side of (2) is one. Nevertheless the squared distance $1+\theta^2/C^2$ generates $F$. The obstruction is therefore present for arbitrarily large actual squared-distance fields, not merely for a field enlarged by irrelevant elements. Totally real fields have unbounded degree, for example multiquadratic fields.

This construction does not refute Problem 132: it only refutes a proposed lower bound on the number of conjugate diameter classes based on field degree and set size alone.

Exact algebraization and the remaining gap

Label any finite planar configuration by $1,\ldots,n$ and introduce real variables $x_i,y_i$. Set

Require $Q_{ij}>0$ for every distinct pair of labels. For every two unordered pairs, require $Q_{ij}=Q_{kl}$ or $Q_{ij}\ne Q_{kl}$ according to the original configuration. These are finitely many polynomial conditions over $\mathbb Q$, with a real solution. Tarski's transfer principle gives a solution over the real algebraic numbers. Its coordinates generate a finite extension of $\mathbb Q$, and its distance partition, hence $R(A)$, is exactly the original one. Numerical lengths need not be preserved.

For each fixed $n$, a counterexample to the two-distance assertion would therefore have an algebraic realization. Likewise, a sequence refuting the growth assertion could be replaced term by term with algebraic realizations. A uniform result over all number fields would suffice for the original questions. This deduction uses the external real-closed-field transfer theorem, rather than a claim that arbitrary real coordinates already belong to a number field.

What is missing is a way to derive rare classes from the preserved pattern without changing it. Algebraization supplies no analog of $s^2\notin F$ for the two coordinate directions, and the conjugate-diameter construction shows that field degree alone cannot supply the missing classes. The surviving arithmetic question is how simultaneous relations among squared differences constrain their multiplicities in configurations that admit only similarities as equality-preserving affine maps. No bound for that remaining class is proved here.

Finite checks and sources

The exact finite probe enumerates all 65,535 nonempty subsets of $\{0,1,2,3\}^2$, applies $(x,y)\mapsto(x,(1+\sqrt2)y)$, and counts squared distances by their integer coefficient pairs in $\mathbb Q(\sqrt2)$. It checks (1), the midline-sensitive bound used in its proof, the separated-displacement conclusion for the biquadratic shear, and the complete seven-point spectrum above. The finite enumeration checks the proposed mechanisms on that domain; the general conclusions rely on the written arguments. It is not an independent review.

Run from the repository root:

uv run --no-sync python wiki/research/leads/number_field_rare_distances/evidence/main.py

Relevant external inputs and their reading scope:

• David Marker, Model Theory for Algebra and Algebraic Geometry, lecture notes, Corollary 7.15: completeness of real closed fields and the elementary inclusion of the real algebraic numbers in $\mathbb R$. The statement and its short displayed deduction were read in indexed text; direct PDF access failed. Its underlying quantifier-elimination proof was not reconstructed. The polynomial-system application above is recorded here.
• Clemen, Dumitrescu, and Liu, On multiplicities of interpoint distances, arXiv:2505.04283v2, Section 1.1: the $n\ge5$ formulation, the classical diameter bound, and Theorems 1.2 and 1.3 were read as statements in the HTML source. The [[../library/distance_problems/clemen_2025_multiplicities_interpoint_distances/_index|library digest]] provides the canonical local source home. No complete proof review of those external results is claimed.
• Alon et al., Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695v1, Section 2: its norm-one and window construction motivates the arithmetic comparison. The [[../library/discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/_index|compiled source]] records its proof coverage. Its lower bound for one rich distance is not a premise of (1).
• Lee, Pohoata, and Zhu, The Minkowski grid has robustly many repeated distances, arXiv:2607.05374v1: the abstract was screened in the status search. Its robust lower bound for a most frequent distance in subsets is a different question; no theorem or proof from it is consumed here.