Planar distinct distances and unit-distance bounds

1 · The essentials

Place any number of distinct points on a flat plane, and choose one as your starting point. Measure the distances from it to all the other points, counting repeated lengths only once. The main manuscript says that, when the number of points is large, almost every starting point gives many different lengths. More precisely, for any fixed positive ε, almost every point in an n-point set gives at least $n^{1 - ε}$ different nonzero distances. Here “almost every” means that the fraction of points allowed to fail tends to zero as the set grows. This main result has a Lean formalization. A companion manuscript addresses a different question: how many pairs of points can be exactly one unit apart, and gives a bound with an exponent strictly below $4/3$ .

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine placing markers on a tabletop and making a list of the different ruler readings from each marker to the others. The main result concerns how long nearly every marker’s list must be, while the companion result counts only pairs whose ruler reading is exactly one chosen unit.

Why it matters

The main result controls distances seen from individual points, rather than merely collecting lengths from the whole set. Its guarantee applies to almost all points in any planar configuration, subject to the stated asymptotic allowance for exceptions. The companion result complements this by limiting how often one particular length can occur.

2 · Going further

For a set $P$ of $n$ points in the Euclidean plane and a point $p \in P$ , consider the set of lengths ${∥ p - q ∥ : q \in P, q \neq = p}$ . The pinned question asks how large this set must be when the starting point $p$ is fixed. The main manuscript states that, for every fixed $ε > 0$ , all but $o (n)$ points $p$ determine at least $n^{1 - ε}$ distinct nonzero distances. The notation $o (n)$ means that the exceptional count divided by $n$ tends to zero.

The source identifies this as the weak pinned Erdős distinct-distance conjecture and reports a Lean formalization of the main result. The supplied abstract states the conclusion but gives no argument or proof mechanism, so no account of the method can be drawn from it.

The companion manuscript shows in the manuscript that there is an absolute exponent $β < 4/3$ for which every $n$ -point planar set determines $O (n^{β})$ unordered unit-distance pairs. Equivalently, writing $δ = 4/3 - β > 0$ , this is the repository summary’s bound $O (n^{4/3 - δ})$ . Here $O (n^{β})$ denotes an upper bound by a constant multiple of $n^{β}$ . The two statements concern the same kind of point sets but count different objects: distinct lengths from a fixed point versus pairs realizing one fixed length. The supplied abstracts do not specify an argument linking them.

Useful notions

Finite sets of points in the Euclidean plane
Distance between two points
Counting different lengths versus counting pairs
Powers and asymptotic growth

What was known before / what is new

Before

The source presents the main question under the name “weak pinned Erdős distinct-distance conjecture.” It also identifies the planar unit-distance problem as the subject of the companion manuscript. No earlier bounds, chronology, or argument methods are supplied.

New

The main result gives the lower bound $n^{1 - ε}$ on the number of distinct nonzero distances from all but $o (n)$ points, for every fixed $ε > 0$ . This result has a Lean formalization. The companion manuscript shows in the manuscript an upper bound for unit-distance pairs with an absolute exponent strictly below $4/3$ .

Limits and assumptions

Both statements concern finite sets in the Euclidean plane. In the main statement, ε is fixed and positive, and the exceptional fraction tends to zero as n grows; the statement does not guarantee the bound at every point or give a fixed positive fraction of n distinct distances. The companion statement asserts the existence of an absolute β below 4/3 but supplies no numerical value. Lean verification is reported for the main pinned-distance result, not for the companion unit-distance claim. The supplied material gives neither proof details nor an explicit list of remaining open questions.

Mini glossary

Pinned distances: The different distances measured from one fixed point to the other points of a set.
Distinct nonzero distances: Different positive lengths, with each length counted only once regardless of how many points realize it.
o(n): A quantity whose ratio to n tends to zero as n grows.
Unordered unit-distance pair: Two points exactly one unit apart, counted without distinguishing which point is listed first.
Power saving: In the companion statement, a reduction of the exponent in the unit-distance upper bound from 4/3 to 4/3 − δ for an absolute δ > 0.

Keywords : Pinned distances · Unit distances · Planar combinatorics

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $o (n)$ points of any n-point planar set determine at least $n^{1 - ε}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O (n^{4/3 - δ})$ for an absolute δ > 0.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/167.md

Manuscripts in this family

★

The weak pinned planar distance theorem

23/09/2026

Verified by Lean

We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but $o (n)$ points of any n-point planar set determine at least $n^{1 - ε}$ distinct nonzero distances.

PDF GitHub folder Lean formalization

A power saving for planar unit distances

23/09/2026

To be verified

We prove a power saving for the planar unit-distance problem: for some absolute $β < 4/3$ , every set of n points in the Euclidean plane determines $O (n^{β})$ unordered pairs at unit distance.

PDF GitHub folder