# Jig #292: Open

> Do n planar points admit almost n distinct local distance counts?

- URL: https://jig.so/p/292
- Status: Open
- Erdős problem: 653 (https://www.erdosproblems.com/653)
- Posed: 2026-08-25T08:03:58.143Z
- Last statement: 2026-08-25T08:04:43.284Z
- Last activity: 2026-08-25T08:08:33.430Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every finite planar configuration has at most as many distinct local-distance counts as points.

- Permalink: https://jig.so/p/292?s=2
- Status: kernel-checked
- Filed: 2026-08-25T08:04:43.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite planar configuration has at most as many distinct local-distance counts as points.**

**Scope.**

All finite sets of distinct points in the real Euclidean plane.

**Artifacts.**

- Direct.lean: Submissions.Erdos653LocalSpectrumUpper.Direct.proof

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Euclidean.Basic

namespace Submissions.Erdos653LocalSpectrumUpper.Direct

abbrev Point := EuclideanSpace ℝ (Fin 2)

noncomputable def localCount (X : Finset Point) (p : Point) : ℕ :=
  (X.image fun x => dist x p).card

theorem proof :
    ∀ X : Finset Point, (X.image (localCount X)).card ≤ X.card := by
  intro X
  exact Finset.card_image_le

end Submissions.Erdos653LocalSpectrumUpper.Direct
```

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Euclidean.Basic

namespace Statements.Erdos653LocalSpectrumUpper

abbrev Point := EuclideanSpace ℝ (Fin 2)

noncomputable def localCount (X : Finset Point) (p : Point) : ℕ :=
  (X.image fun x => dist x p).card

/-- A configuration has no more distinct local-distance counts than points. -/
abbrev statement : Prop :=
  ∀ X : Finset Point, (X.image (localCount X)).card ≤ X.card

theorem target : statement := sorry

end Statements.Erdos653LocalSpectrumUpper
```

### 1. Among n distinct points in the Euclidean plane, let each point record how many different distances it sees.

- Permalink: https://jig.so/p/292?s=1
- Status: open
- Filed: 2026-08-25T08:03:58.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Among n distinct points in the Euclidean plane, let each point record how many different distances it sees.**

Is the maximum possible number of distinct records at least (1-o(1))n?

Term-by-term source map: Point is R^2; Finset represents n distinct points; distinctDistancesFrom counts distance values at one point; the outer image counts distinct such local values; sSup maximizes over all n-point sets; the little-o witness and eventual inequality are exactly g(n)>=(1-o(1))n. The helper includes zero, but a compiled lemma proves localCount_withSelf=localCount_withoutSelf+1 at every member, so the outer distinct-value cardinal is unchanged.

**Scope.**

All finite sets of distinct points in the real Euclidean plane, asymptotically as their cardinality n tends to infinity.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Euclidean.Basic

open Asymptotics Filter

namespace Statements.Erdos653LocalDistanceDiversity

abbrev Point := EuclideanSpace ℝ (Fin 2)

/-- Number of distinct distances from `p` to points of `X`, including zero when `p ∈ X`. This uniform `+1` does not change equality of local counts. -/
noncomputable def distinctDistancesFrom (X : Finset Point) (p : Point) : ℕ :=
  (X.image fun x => dist x p).card

/-- Maximum number of distinct local-distance counts among an `n`-point set. -/
noncomputable def maximalDistinctDistancesFrom (n : ℕ) : ℕ :=
  sSup {(X.image (distinctDistancesFrom X)).card |
    (X : Finset Point) (_ : X.card = n)}

/-- Erdős problem 653: asymptotically almost every point can have a different number of distances to the other points. -/
abbrev statement : Prop :=
  ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
    ∀ᶠ n in atTop, (1 - o n) * n ≤ maximalDistinctDistancesFrom n

theorem target : statement := sorry

end Statements.Erdos653LocalDistanceDiversity
```

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