Distances from one point and limits on repeated unit distances

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Family 167 · Combinatorics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Stand at one point in a finite planar configuration and measure the distances to all the others. How many different lengths must you see? Now ask a complementary question: how many pairs in the entire configuration can have one particular length, say one?

Family 167 contains two manuscripts addressing these questions. M1 claims that almost every point of an arbitrary large planar set sees at least $n^{1-\varepsilon}$ distances, for any fixed positive $\varepsilon$. M2 claims that the number of unit-distance pairs is bounded by a power strictly smaller than $n^{4/3}$.

The first is a statement about distance diversity at individual locations, called pins. The second controls repetitions of one chosen length across the whole set. Their conclusions have different quantifiers and neither should be treated as a direct restatement of the other.

2. Background and essential definitions

For a finite set $P\subset\mathbb R^2$ and a pin $x\in P$, define

$$D_x(P)=\{\|y-x\|:y\in P\setminus\{x\}\}.$$

The global distance set is the union of these sets over all pins. A large union does not automatically mean that one particular pin sees equally many distances; information may be spread among different locations.

For a pair $x\ne y$, define its distance-fiber size

$$k_P(x,y)= \bigl|\{z\in P\setminus\{x\}:\|z-x\|=\|y-x\|\}\bigr|.$$

This counts the points on the circle centered at $x$ through $y$, including $y$. Every pin uses its own circles. Pairs with the same numerical length but different centers do not belong to one common fiber.

For the second problem, let $u(P)$ count unordered distinct pairs at distance one, and let $u(n)$ be the maximum of $u(P)$ over all $n$-point planar sets. Rescaling converts any fixed positive distance into one, so this is a universal repeated-distance problem.

The notation $o(n)$ means a quantity whose ratio to $n$ tends to zero. Thus “all but $o(n)$ pins” permits exceptional points but requires their proportion to vanish. A power saving in an upper bound replaces an exponent $a$ by $a-\delta$ for a fixed $\delta>0$; it is stronger than merely dividing $n^a$ by a logarithm.

3. Development history and previous partial results

Erdős introduced the planar distance questions in 1946 and explicitly posed a weak pinned question in 1957. The global and pinned problems developed along related but distinct tracks.

Guth and Katz proved that every planar $n$-point set determines at least a constant multiple of $n/\log n$ distances in total. For pinned distances, Solymosi and Tóth obtained an exponent $6/7$, and Tardos and then Katz–Tardos improved it using entropy and geometric counting. M1 records an exponent approaching approximately 0.864137 from that line of work.

The weak pinned conjecture asks for every exponent below one. A sharper version asks for a pin seeing order $n/\sqrt{\log n}$ distances, matching the scale suggested by the square grid. The new claim addresses the weak version and strengthens its one-pin conclusion to almost every pin.

For unit distances, Spencer, Szemerédi and Trotter established $u(n)=O(n^{4/3})$ in 1984. Székely later gave an influential crossing-number proof by viewing the problem as incidences between points and unit circles. Refinements improved constants and examined near-extremal structure, but a fixed improvement of the exponent required additional information.

The recent lower-bound history matters here. M2 cites the May 2026 OpenAI construction of many unit distances, which gives a fixed exponent above one, and Sawin’s refinement, giving exponent 1.014114 along an unbounded sequence of sizes. These are construction lower bounds. The new upper-bound claim leaves a gap above them and does not restore the older proposed $n^{1+o(1)}$ upper bound.

4. What the new manuscripts claim

M1: large distance fibers occupy a vanishing fraction of pairs

For every fixed $s>0$, M1’s Theorem 1.1 states

$$\sup_{|P|=n} \frac{\bigl|\{(x,y):x\ne y,\ k_P(x,y)\ge n^s\}\bigr|} {n(n-1)}\longrightarrow0.$$

The supremum ranges over all planar configurations of $n$ distinct points. The convergence is uniform over those configurations, with no separation or general-position assumption.

Corollary 1.2 says that for every fixed $\varepsilon>0$, all but $o(n)$ pins satisfy

$$|D_x(P)|\ge n^{1-\varepsilon}.$$

The source supplies no explicit rate for the exceptional fraction. It cannot exclude every bad pin: points on a circle together with its center give a center that sees only one distance.

M2: a fixed improvement for unit distances

M2’s Theorem 1.1 asserts absolute constants $C<\infty$ and $1\le\beta<4/3$ such that

$$u(n)\le Cn^\beta \qquad\text{for every }n\ge0.$$

Equivalently, the exponent is $4/3-\delta$ for some absolute $\delta>0$. The count is of unordered pairs. The constants do not depend on the configuration, and no numerical value for the saving is extracted.

These are separate manuscripts with different conclusions. The fiber theorem controls a vanishing proportion of pairs at each fixed power threshold; it does not automatically produce M2’s fixed power saving for one repeated length.

5. Why the advance matters

Importance: 4/5. Both claims would break longstanding barriers in planar distance geometry. The pinned theorem approaches linear distance diversity and strengthens existence of one good pin to a statement about almost every location.

The unit-distance result would improve the exponent governing one of the basic incidence problems. Reducing a fixed exponent is a stronger asymptotic change than a constant improvement and calls for structure beyond the standard point–circle incidence estimate.

The rating reflects major advances while retaining the unresolved quantitative gaps. M1 does not reach the sharp pinned scale, and M2 does not determine the optimal unit-distance exponent. Their methods combine arithmetic and probability with Euclidean geometry in a way that may be useful beyond the headline bounds.

6. The main idea

Factor distance in complex coordinates

Both sources use the coordinates

$$z=x_1+ix_2,\qquad w=x_1-ix_2.$$

For two real planar points,

$$\|x-y\|^2=(z_x-z_y)(w_x-w_y).$$

On a fixed circle around a pin, this product is constant. On a unit edge it equals one, so the two coordinate differences are reciprocals. This converts geometric constraints into multiplicative algebraic relations.

The sources transfer finite systems of distance equalities and inequalities to algebraic coordinates. They then use the product formula for number fields, which balances logarithmic absolute values across all real, complex and prime-adic measurements. A difference that is small in one measurement cannot be treated in isolation from the others.

M1: compare a distance fiber with the whole set

Assuming many pairs lie in large fibers, M1 organizes points into nested partitions according to coordinate closeness at each absolute value. Their overlap counts satisfy an additive identity derived from the product formula.

The source compares sampling uniformly from a fiber with sampling uniformly from the whole configuration. The identity turns the comparison into variance bounds: the two distributions must agree unusually well across many partition cells.

Two scale regimes remain. If the integrated overlap scale grows without bound, the nested partitions behave like weighted trees, and the required concentration of tree distances contradicts their geometry. If the scale stays bounded, normalized coordinate distributions converge to nonatomic limits, meaning no single point has positive mass.

The circle relation becomes a family of fractional linear, or Möbius, maps between those limiting distributions. The independence and variance properties force an incompatible collapse of their spread. This rules out a positive density of large-fiber pairs.

The almost-every-pin deduction is simpler: a pin seeing too few distances must place most of its neighbors into large fibers, so many bad pins would contradict the pair-density theorem.

M2: predict neighbors and extract an algebraic obstruction

M2 begins with a hypothetical sequence approaching the $4/3$ exponent. It restricts the point–circle incidence graph to a part with controlled degrees and sampling laws. Entropy estimates show that two points sampled as neighbors of one center share relatively little information.

A short history of coordinate tests can then predict enough about a point to preserve its compatibility with a fresh neighboring center. Arithmetic height measures the total logarithmic size of algebraic quantities across absolute values.

In a bounded-height regime, predictions force a separated point to approximately satisfy too many unit-circle equations, producing a geometric contradiction. In the unbounded regime, the source extracts ratios that individually have large height while certain rectangular products have negligible height.

Passing to a limiting field converts that separation into algebraic independence. A system of quadratic extensions then permits independent sign changes of square roots that conflict with the required equations. The argument excludes every sequence approaching exponent $4/3$, yielding some fixed gap without estimating its size.

7. Scope, limitations, and open questions

M1 fixes $\varepsilon$ or $s$ before taking the large-set limit. It does not provide one explicit exceptional-count formula valid for parameters shrinking with $n$. Exceptional pins are unavoidable in general.

M2 gives existence of an absolute exponent improvement, with no numerical value for $\delta$. It does not establish a near-linear upper bound or match the recent lower constructions. The maximum possible exponent remains undetermined within these claims.

Both results concern finite sets of distinct points in the real Euclidean plane. They do not directly apply to arbitrary norms, finite-field planes or configurations with repeated points.

The global, pinned and repeated-distance statistics should remain separate when applying these results. Each asks a different question about how equal lengths are distributed, even though the same coordinate factorization helps study them.

8. Sources and further reading

M1’s Theorem 1.1 and Corollary 1.2 give the fiber and pinned statements; Sections 3–7 explain the partition identity and two scale regimes. M2’s Theorem 1.1 gives the unit-distance saving; its introduction outlines prediction, height extraction and the final algebraic obstruction.

The primary historical links above distinguish global distance bounds from unit-distance constructions. Wikipedia’s unit distance problem and mutual information pages provide optional background.

Source manuscripts

• M1. OpenAI, The weak pinned planar distance theorem (September 23, 2026).
• M2. OpenAI, A power saving for planar unit distances (September 23, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.