The higher-dimensional Erdős distinct-distances conjecture

1 · The essentials

Take a collection of different points and measure the distance between every pair. Many pairs may have the same distance, so the question is how many different distance values must appear. The manuscript claims an answer for every fixed dimension of three or more. For any collection of at least two points, it gives a minimum number of different distances that grows with the number of points. The claimed minimum is a positive constant times the number of points raised to the power two divided by the dimension. The repository summary says this matches the growth found in an integer grid, up to a constant factor. The main result is not formalized in Lean and has not been formally verified.

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

As an imperfect image, imagine placing beads at different positions in a room and making a list of all the bead-to-bead lengths, writing each length only once. The claim says that, however you arrange the beads, that list cannot stay too short as you add more beads; the room image covers only the three-dimensional case.

Why it matters

The manuscript describes its claim as a positive resolution of the higher-dimensional Erdős distinct-distances conjecture. According to the repository summary, the bound matches the integer-grid order up to a constant factor, rather than merely giving a lower bound with a smaller power of the number of points.

2 · Going further

For a finite set $P \subset R^{d}$ of $n$ distinct points, let $D (P)$ be the set of Euclidean distances between different points of $P$ . The precise claim is that, for every fixed integer $d \geq 3$ , there is a constant $c_{d} > 0$ such that $∣ D (P) ∣ \geq c_{d} n^{2/ d}$ for every $n \geq 2$ and every such set $P$ . The constant depends only on the dimension, not on the number or arrangement of the points.

The manuscript identifies this statement as the higher-dimensional Erdős distinct-distances conjecture and claims to resolve it positively. The repository summary adds that the exponent $2/ d$ matches the integer-grid order, making the claimed bound sharp in its growth rate up to a constant factor. This is not a claim about an exact minimum or an optimal value of $c_{d}$ .

The supplied abstract states the result but gives no argument, intermediate estimates, or description of a method. Consequently, the main idea of the argument cannot be explained from this source alone. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described.

Useful notions

Points and Euclidean distance
Counting different values rather than pairs
Coordinates and dimension
Powers and growth rates
Lower bounds and constant factors

What was known before / what is new

Before

The source presents the target statement as the higher-dimensional Erdős distinct-distances conjecture. The repository summary identifies the integer-grid order as the benchmark for the claimed bound. No earlier lower bounds or historical development are supplied.

New

The manuscript claims the bound $c_{d} n^{2/ d}$ for every fixed integer dimension $d \geq 3$ and every set of $n \geq 2$ distinct points in $R^{d}$ . The claimed constant is positive and depends only on $d$ . The repository summary describes this as a constant-factor resolution matching the integer-grid order.

Limits and assumptions

The claim concerns distinct points in Euclidean space, with a fixed integer dimension $d \geq 3$ and at least two points. It does not provide a dimension-independent constant, and the supplied text gives no explicit value of $c_{d}$ or argument supporting the bound. Dimensions below three are outside the stated result. The main result is not formalized in Lean and has not been formally verified. The source identifies no remaining open questions; its assertion that the conjecture is resolved is the manuscript's claim.

Mini glossary

Distinct distances: The different distance values occurring between pairs of points, with repeated values counted only once.
Euclidean space: The coordinate space $R^{d}$ equipped with ordinary straight-line distance.
Lower bound: A quantity that the number under consideration is asserted to be at least.
Constant-factor bound: A bound that matches a stated growth rate apart from a fixed multiplicative constant.
Integer grid: An arrangement of points whose coordinates are integers.

Keywords : Distinct distances · Erdős conjecture · Euclidean geometry

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)

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝ^d determine at least $c_{d} n^{2/ d}$ distinct distances, with $c_{d} > 0$ depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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The higher-dimensional Erdős distinct-distances conjecture

23/09/2026

To be verified

For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝ^d determines at least $c_{d} n^{2/ d}$ distinct distances, where $c_{d} > 0$ depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.

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