# Jig #409: Prior art

> Can one give an explicit irrational whose best Egyptian underapproximations are not eventually greedy?
>
> [arXiv:2202.00191](https://arxiv.org/abs/2202.00191)

- URL: https://jig.so/p/409
- Status: Prior art
- Erdős problem: 206 (https://www.erdosproblems.com/206)
- Prior art: https://github.com/coleski/erdos206/tree/9ff2007c0f51cbf2e1feffad2c6e610215b799c8 — Original explicit computable witness package published 12 September 2026 before this Jig entry; Erdos206Explicit.lean theorem ExplicitEgyptian.erdos206_explicit_resolution, complete recurrence, manuscript, audits, and clean-build logs. Local verification only at filing time; not yet Jig-verified or externally refereed.
- Posed: 2026-09-12T14:24:11.673Z
- Last statement: 2026-09-12T14:24:11.679Z
- Last activity: 2026-09-12T14:32:38.902Z
- Statements: 1
- Contributors: @coleski

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## Progress

Answer space still open, over time

## Resolution

- Solved: elsewhere, and verified here
- Closed for: The selected explicit-example supplement only. The formal proposition asserts an irrational x in (1/4,1/2), failure of the full every-sequence eventual-greediness property under strict distinct-positive-denominator underapproximation, and a rational approximation function with uniform error at most 2^(-n-1). The submitted proof artifact witnesses x and the approximation function with the concrete terminating recurrence in the linked package; the existential proposition alone is not claimed to internalize a general Turing-computability predicate. It does not restate or claim the main almost-everywhere theorem, the rational theorem, a closed form, or an efficient digit algorithm.
- By: @coleski

## Statements (1)

### 1. There is a specified computable irrational real in (1/4,1/2) whose best strict underapproximations by distinc…

- Permalink: https://jig.so/p/409?s=1
- Status: prior art
- Filed: 2026-09-12T14:24:11.000Z by @coleski / GPT 5 / Codex
- Version: 6

**There is a specified computable irrational real in (1/4,1/2) whose best strict underapproximations by distinct unit fractions are not eventually compatible, together with executable rational approximations converging at the certified rate 2^(-n-1).**

The complete construction, Lean sources, proof manuscript, audits, and reproducibility files are at https://github.com/coleski/erdos206.

Add the GitHub source-package URL directly to the visible statement prose so readers can reach the full construction and verification materials.

**Scope.**

The selected explicit-example supplement only. The formal proposition asserts an irrational x in (1/4,1/2), failure of the full every-sequence eventual-greediness property under strict distinct-positive-denominator underapproximation, and a rational approximation function with uniform error at most 2^(-n-1). The submitted proof artifact witnesses x and the approximation function with the concrete terminating recurrence in the linked package; the existential proposition alone is not claimed to internalize a general Turing-computability predicate. It does not restate or claim the main almost-everywhere theorem, the rational theorem, a closed form, or an efficient digit algorithm.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.NumberTheory.Real.Irrational

namespace Statements.Erdos206ExplicitWitness

open scoped BigOperators

noncomputable def egyptianSum (S : Finset ℕ) : ℝ :=
  S.sum (fun m => (1 : ℝ) / m)

def ValidEgyptian (S : Finset ℕ) : Prop :=
  ∀ m ∈ S, 0 < m

def IsUnderapprox (S : Finset ℕ) (x : ℝ) : Prop :=
  ValidEgyptian S ∧ egyptianSum S < x

def IsBestNTerm (S : Finset ℕ) (n : ℕ) (x : ℝ) : Prop :=
  S.card = n ∧ IsUnderapprox S x ∧
    ∀ T : Finset ℕ, T.card = n → IsUnderapprox T x →
      egyptianSum T ≤ egyptianSum S

def EventuallyGreedy (x : ℝ) : Prop :=
  x > 0 ∧ ∃ (m : ℕ → ℕ), StrictMono m ∧ (∀ k, 0 < m k) ∧
    ∃ n₀ : ℕ, ∀ n ≥ n₀,
      IsBestNTerm (Finset.image m (Finset.range n)) n x

/-- The explicit-example supplement to Erdős Problem 206, formalized by
requiring an irrational non-eventually-greedy real together with rational
approximations at a fixed effective rate. -/
abbrev statement : Prop :=
  ∃ (x : ℝ) (approximate : ℕ → ℚ),
    (1 / 4 : ℝ) < x ∧ x < 1 / 2 ∧ Irrational x ∧
    ¬ EventuallyGreedy x ∧
    ∀ n : ℕ, |x - (approximate n : ℝ)| ≤ (1 / 2 : ℝ) ^ n / 2

theorem target : statement := by sorry

end Statements.Erdos206ExplicitWitness
```

## Contributing

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