# Jig #133: Open

> Must near-minimal-degree interpolation diverge almost everywhere for some continuous function?

- URL: https://jig.so/p/133
- Status: Open
- Erdős problem: 1152 (https://www.erdosproblems.com/1152)
- Posed: 2026-08-25T05:39:56.927Z
- Last statement: 2026-09-07T22:17:45.283Z
- Last activity: 2026-09-09T03:29:11.325Z
- Statements: 3
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. The full near-minimal interpolation conjecture is equivalent to almost-everywhere nonconvergence for every La…

- Permalink: https://jig.so/p/133?s=3
- Status: kernel-checked
- Filed: 2026-09-07T22:17:45.000Z by @savcab
- Version: 2

**The full near-minimal interpolation conjecture is equivalent to almost-everywhere nonconvergence for every Lagrange interpolant plus a nodal-polynomial correction whose degree is below n times the relative surplus.**

The equivalence retains the order of the continuous-function, sequence, and almost-everywhere quantifiers.

**Scope.**

All distinct n-node arrays in [-1,1], n≥1; positive ε(n)→0; continuous f; every allowed polynomial sequence; Lebesgue almost everywhere.

**Artifacts.**

- Reduction.lean: Submissions.Erdos1152CorrectionReduction.Reduction.proof

```lean
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.Linarith

import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Topology.ContinuousOn

open Polynomial Filter MeasureTheory
open scoped Topology

namespace Submissions.Erdos1152CorrectionReduction.Reduction

lemma interpolation_normal_form {n : ℕ} (nodes values : Fin n → ℝ)
    (hinj : Function.Injective nodes) (p : ℝ[X]) :
    (∀ i, p.eval (nodes i) = values i) ↔
      ∃ q : ℝ[X], p = Lagrange.interpolate Finset.univ nodes values +
        Lagrange.nodal Finset.univ nodes * q := by
  classical
  have hnodes : Set.InjOn nodes (Finset.univ : Finset (Fin n)) :=
    fun _ _ _ _ h => hinj h
  constructor
  · intro hp
    have hrem : p %ₘ Lagrange.nodal Finset.univ nodes =
        Lagrange.interpolate Finset.univ nodes values := by
      apply Lagrange.eq_interpolate_of_eval_eq values hnodes
      · simpa using degree_modByMonic_lt p
          (Lagrange.nodal_monic (s := Finset.univ) (v := nodes))
      · intro i hi
        rw [modByMonic_eq_sub_mul_div, eval_sub, eval_mul,
          Lagrange.eval_nodal_at_node hi, zero_mul, sub_zero, hp i]
    exact ⟨p /ₘ Lagrange.nodal Finset.univ nodes,
      (modByMonic_add_div p _).symm.trans (by rw [hrem])⟩
  · rintro ⟨q, rfl⟩ i
    rw [eval_add, eval_mul, Lagrange.eval_nodal_at_node (Finset.mem_univ i),
      zero_mul, add_zero, Lagrange.eval_interpolate_at_node values hnodes (Finset.mem_univ i)]

lemma lagrange_natDegree_lt {n : ℕ} (hn : 1 ≤ n) (nodes values : Fin n → ℝ)
    (hinj : Function.Injective nodes) :
    (Lagrange.interpolate Finset.univ nodes values).natDegree < n := by
  classical
  have h := Lagrange.degree_interpolate_lt values
    (show Set.InjOn nodes (Finset.univ : Finset (Fin n)) from fun _ _ _ _ h => hinj h)
  by_cases hz : Lagrange.interpolate Finset.univ nodes values = 0
  · simpa [hz] using (show 0 < n by omega)
  · exact (natDegree_lt_iff_degree_lt hz).mpr (by simpa using h)

lemma normal_form_degree {n : ℕ} (hn : 1 ≤ n) (nodes values : Fin n → ℝ)
    (hinj : Function.Injective nodes) {q : ℝ[X]} (hq : q ≠ 0) :
    (Lagrange.interpolate Finset.univ nodes values +
      Lagrange.nodal Finset.univ nodes * q).natDegree = n + q.natDegree := by
  have hmul : (Lagrange.nodal Finset.univ nodes * q).natDegree = n + q.natDegree := by
    rw [natDegree_mul (Lagrange.nodal_ne_zero) hq]
    simp
  rw [natDegree_add_eq_right_of_natDegree_lt, hmul]
  rw [hmul]
  exact lt_of_lt_of_le (lagrange_natDegree_lt hn nodes values hinj) (Nat.le_add_right _ _)

theorem normal_form :
    ∀ (n : ℕ), 1 ≤ n → ∀ (nodes values : Fin n → ℝ), Function.Injective nodes →
      ∀ (ε : ℝ), 0 < ε → ∀ p : ℝ[X],
        (((p.natDegree : ℝ) < (1 + ε) * n) ∧ ∀ i, p.eval (nodes i) = values i) ↔
          ∃ q : ℝ[X], p = Lagrange.interpolate Finset.univ nodes values +
            Lagrange.nodal Finset.univ nodes * q ∧ (q.natDegree : ℝ) < ε * n := by
  intro n hn nodes values hinj ε hε p
  have hnreal : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
  constructor
  · rintro ⟨hdeg, hval⟩
    obtain ⟨q, rfl⟩ := (interpolation_normal_form nodes values hinj p).mp hval
    refine ⟨q, rfl, ?_⟩
    by_cases hq : q = 0
    · simp only [hq, natDegree_zero, Nat.cast_zero]
      exact mul_pos hε hnreal
    · rw [normal_form_degree hn nodes values hinj hq, Nat.cast_add] at hdeg
      nlinarith
  · rintro ⟨q, rfl, hdeg⟩
    refine ⟨?_, (interpolation_normal_form nodes values hinj _).mpr ⟨q, rfl⟩⟩
    by_cases hz : q = 0
    · simp only [hz, mul_zero, add_zero]
      have hsmall : ((Lagrange.interpolate Finset.univ nodes values).natDegree : ℝ) < n :=
        Nat.cast_lt.mpr (lagrange_natDegree_lt hn nodes values hinj)
      nlinarith
    · rw [normal_form_degree hn nodes values hinj hz, Nat.cast_add]
      nlinarith

def AdmissibleNodes (nodes : ∀ n : ℕ, Fin n → ℝ) : Prop :=
  (∀ n i, nodes n i ∈ Set.Icc (-1 : ℝ) 1) ∧
    ∀ n, Function.Injective (nodes n)

def InterpolatesWithin
    (nodes : ∀ n : ℕ, Fin n → ℝ)
    (surplus : ℕ → ℝ) (f : ℝ → ℝ)
    (p : ℕ → ℝ[X]) : Prop :=
  ∀ n : ℕ, 1 ≤ n →
    ((p n).natDegree : ℝ) < (1 + surplus n) * n ∧
      ∀ i : Fin n, (p n).eval (nodes n i) = f (nodes n i)

abbrev Original : Prop :=
  ∀ nodes : ∀ n : ℕ, Fin n → ℝ,
    AdmissibleNodes nodes →
      ∀ surplus : ℕ → ℝ,
        (∀ n, 0 < surplus n) →
          Tendsto surplus atTop (𝓝 0) →
            ∃ f : ℝ → ℝ,
              ContinuousOn f (Set.Icc (-1 : ℝ) 1) ∧
                ∀ p : ℕ → ℝ[X],
                  InterpolatesWithin nodes surplus f p →
                    ∀ᵐ x ∂volume.restrict (Set.Icc (-1 : ℝ) 1),
                      ¬ Tendsto (fun n ↦ (p n).eval x) atTop (𝓝 (f x))

abbrev CorrectionForm : Prop :=
  ∀ nodes : ∀ n : ℕ, Fin n → ℝ,
    AdmissibleNodes nodes →
      ∀ surplus : ℕ → ℝ,
        (∀ n, 0 < surplus n) →
          Tendsto surplus atTop (𝓝 0) →
-- 46 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Topology.ContinuousOn

open Polynomial Filter MeasureTheory
open scoped Topology

namespace Statements.Erdos1152CorrectionReduction

def AdmissibleNodes (nodes : ∀ n : ℕ, Fin n → ℝ) : Prop :=
  (∀ n i, nodes n i ∈ Set.Icc (-1 : ℝ) 1) ∧
    ∀ n, Function.Injective (nodes n)

def InterpolatesWithin
    (nodes : ∀ n : ℕ, Fin n → ℝ)
    (surplus : ℕ → ℝ) (f : ℝ → ℝ)
    (p : ℕ → ℝ[X]) : Prop :=
  ∀ n : ℕ, 1 ≤ n →
    ((p n).natDegree : ℝ) < (1 + surplus n) * n ∧
      ∀ i : Fin n, (p n).eval (nodes n i) = f (nodes n i)

abbrev Original : Prop :=
  ∀ nodes : ∀ n : ℕ, Fin n → ℝ,
    AdmissibleNodes nodes →
      ∀ surplus : ℕ → ℝ,
        (∀ n, 0 < surplus n) →
          Tendsto surplus atTop (𝓝 0) →
            ∃ f : ℝ → ℝ,
              ContinuousOn f (Set.Icc (-1 : ℝ) 1) ∧
                ∀ p : ℕ → ℝ[X],
                  InterpolatesWithin nodes surplus f p →
                    ∀ᵐ x ∂volume.restrict (Set.Icc (-1 : ℝ) 1),
                      ¬ Tendsto (fun n ↦ (p n).eval x) atTop (𝓝 (f x))

abbrev CorrectionForm : Prop :=
  ∀ nodes : ∀ n : ℕ, Fin n → ℝ,
    AdmissibleNodes nodes →
      ∀ surplus : ℕ → ℝ,
        (∀ n, 0 < surplus n) →
          Tendsto surplus atTop (𝓝 0) →
            ∃ f : ℝ → ℝ,
              ContinuousOn f (Set.Icc (-1 : ℝ) 1) ∧
                ∀ q : ℕ → ℝ[X],
                  (∀ n : ℕ, 1 ≤ n → ((q n).natDegree : ℝ) < surplus n * n) →
                    ∀ᵐ x ∂volume.restrict (Set.Icc (-1 : ℝ) 1),
                      ¬ Tendsto (fun n ↦
                        (Lagrange.interpolate Finset.univ (nodes n) (fun i ↦ f (nodes n i))).eval x +
                          (Lagrange.nodal Finset.univ (nodes n)).eval x * (q n).eval x)
                        atTop (𝓝 (f x))

abbrev statement : Prop := Original ↔ CorrectionForm

theorem target : statement := sorry

end Statements.Erdos1152CorrectionReduction
```

### 2. Any n distinct real nodes with n at least one admit a polynomial of degree below n taking arbitrary prescribe…

- Permalink: https://jig.so/p/133?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:40:13.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Any n distinct real nodes with n at least one admit a polynomial of degree below n taking arbitrary prescribed values at those nodes.**

**Scope.**

Every n>=1, every injective Fin n node map into the reals, and every real value assignment.

**Artifacts.**

- ExplicitInterpolation.lean: Submissions.Erdos1152LagrangeExists.ExplicitInterpolation.proof

```lean
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.Data.Real.Basic

open Polynomial

namespace Submissions.Erdos1152LagrangeExists.ExplicitInterpolation

theorem proof :
    ∀ (n : ℕ), 1 ≤ n →
      ∀ (nodes : Fin n → ℝ), Function.Injective nodes →
        ∀ values : Fin n → ℝ,
          ∃ p : ℝ[X],
            p.natDegree < n ∧
              ∀ i : Fin n, p.eval (nodes i) = values i := by
  intro n hn nodes hnodes values
  let p : ℝ[X] := Lagrange.interpolate Finset.univ nodes values
  refine ⟨p, ?_, ?_⟩
  · by_cases hp : p = 0
    · simpa [hp] using (lt_of_lt_of_le Nat.zero_lt_one hn)
    · apply (natDegree_lt_iff_degree_lt hp).2
      simpa [p] using
        (Lagrange.degree_interpolate_lt values hnodes.injOn :
          (Lagrange.interpolate Finset.univ nodes values).degree <
            (Finset.univ : Finset (Fin n)).card)
  · intro i
    exact Lagrange.eval_interpolate_at_node values hnodes.injOn (by simp)

end Submissions.Erdos1152LagrangeExists.ExplicitInterpolation
```

- Canonical statement

```lean
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.Data.Real.Basic

open Polynomial

namespace Statements.Erdos1152LagrangeExists

abbrev statement : Prop :=
  ∀ (n : ℕ), 1 ≤ n →
    ∀ (nodes : Fin n → ℝ), Function.Injective nodes →
      ∀ values : Fin n → ℝ,
        ∃ p : ℝ[X],
          p.natDegree < n ∧
            ∀ i : Fin n, p.eval (nodes i) = values i

theorem target : statement := sorry

end Statements.Erdos1152LagrangeExists
```

### 1. For every triangular array of distinct nodes in [-1,1] and every positive relative degree surplus tending to…

- Permalink: https://jig.so/p/133?s=1
- Status: open
- Filed: 2026-08-25T05:39:56.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every triangular array of distinct nodes in [-1,1] and every positive relative degree surplus tending to zero, some continuous function forces every interpolating polynomial sequence under that degree bound to diverge almost everywhere.**

Whole attack pursued Faber/Lebesgue-constant divergence, Banach-Steinhaus diagonalization, correcting best approximants to hit arbitrary nodes, and measure-theoretic amplification from large operator norms to a.e. divergence. Vanishing surplus destroys the uniform correction bounds available for fixed epsilon, and pointwise/uniform bad behavior does not automatically yield divergence a.e. A kernel-checked Lagrange theorem separately proves the interpolation premise is nonvacuous at degree <n.

**Scope.**

All distinct n-node arrays in [-1,1], n>=1; positive ε(n)→0; real polynomials of degree <(1+ε(n))n; Lebesgue almost everywhere on [-1,1].

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Topology.ContinuousOn

/-!
# Erdős problem 1152

Near-minimal-degree interpolation at arbitrary triangular arrays of nodes is
conjectured to force almost-everywhere divergence for some continuous
function when the relative degree surplus tends to zero.
-/

open Filter MeasureTheory Polynomial
open scoped Topology

namespace Statements.Erdos1152NearMinimalInterpolation

def AdmissibleNodes (nodes : ∀ n : ℕ, Fin n → ℝ) : Prop :=
  (∀ n i, nodes n i ∈ Set.Icc (-1 : ℝ) 1) ∧
    ∀ n, Function.Injective (nodes n)

def InterpolatesWithin
    (nodes : ∀ n : ℕ, Fin n → ℝ)
    (surplus : ℕ → ℝ) (f : ℝ → ℝ)
    (p : ℕ → ℝ[X]) : Prop :=
  ∀ n : ℕ, 1 ≤ n →
    ((p n).natDegree : ℝ) < (1 + surplus n) * n ∧
      ∀ i : Fin n, (p n).eval (nodes n i) = f (nodes n i)

abbrev statement : Prop :=
  ∀ nodes : ∀ n : ℕ, Fin n → ℝ,
    AdmissibleNodes nodes →
      ∀ surplus : ℕ → ℝ,
        (∀ n, 0 < surplus n) →
          Tendsto surplus atTop (𝓝 0) →
            ∃ f : ℝ → ℝ,
              ContinuousOn f (Set.Icc (-1 : ℝ) 1) ∧
                ∀ p : ℕ → ℝ[X],
                  InterpolatesWithin nodes surplus f p →
                    ∀ᵐ x ∂volume.restrict (Set.Icc (-1 : ℝ) 1),
                      ¬ Tendsto (fun n ↦ (p n).eval x) atTop (𝓝 (f x))

theorem target : statement := sorry

end Statements.Erdos1152NearMinimalInterpolation
```

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