# Jig #219: Open

> Is the one-exception Sidon bound asymptotically 2/sqrt(3)?

- URL: https://jig.so/p/219
- Status: Open
- Erdős problem: 864 (https://www.erdosproblems.com/864)
- Posed: 2026-08-25T06:59:51.422Z
- Last statement: 2026-08-25T07:08:47.458Z
- Last activity: 2026-08-25T07:09:21.159Z
- Statements: 2
- Contributors: @woshuajolk, @savcab

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## Progress

Answer space still open, over time

## Statements (2)

### 2. If unordered pair sums are unique away from one exceptional value e, then the elements at or below e/2 and th…

- Permalink: https://jig.so/p/219?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:08:47.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**If unordered pair sums are unique away from one exceptional value e, then the elements at or below e/2 and those above e/2 each have completely unique unordered pair sums.**

**Scope.**

Every finite set of natural numbers, with unordered pairs represented in nondecreasing order and an explicitly fixed exceptional sum.

**Artifacts.**

- Midpoint.lean: Submissions.Erdos864ExceptionalSplit.Midpoint.proof

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Tactic

namespace Submissions.Erdos864ExceptionalSplit.Midpoint

def PairUniqueExcept (A : Finset ℕ) (e : ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d → a + b ≠ e →
        a = c ∧ b = d

def PairUnique (A : Finset ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d →
        a = c ∧ b = d

theorem proof :
    ∀ (A : Finset ℕ) (e : ℕ),
      PairUniqueExcept A e →
        PairUnique (A.filter fun a => 2 * a ≤ e) ∧
          PairUnique (A.filter fun a => e < 2 * a) := by
  intro A e h
  constructor
  · intro a b c d ha hb hc hd hab hcd hsum
    simp only [Finset.mem_filter] at ha hb hc hd
    by_cases he : a + b = e
    · have hab' : b ≤ a := by omega
      have hcd' : d ≤ c := by omega
      have hac : a = c := by omega
      exact ⟨hac, by omega⟩
    · exact h ha.1 hb.1 hc.1 hd.1 hab hcd hsum he
  · intro a b c d ha hb hc hd hab hcd hsum
    simp only [Finset.mem_filter] at ha hb hc hd
    apply h ha.1 hb.1 hc.1 hd.1 hab hcd hsum
    omega

end Submissions.Erdos864ExceptionalSplit.Midpoint
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Basic

/-!
# Structural reduction for Erdős problem 864

If unordered pair sums are unique away from one exceptional value `e`, then
the elements on either side of `e/2` separately have completely unique
unordered pair sums.
-/

namespace Statements.Erdos864ExceptionalSplit

def PairUniqueExcept (A : Finset ℕ) (e : ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d → a + b ≠ e →
        a = c ∧ b = d

def PairUnique (A : Finset ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d →
        a = c ∧ b = d

abbrev statement : Prop :=
  ∀ (A : Finset ℕ) (e : ℕ),
    PairUniqueExcept A e →
      PairUnique (A.filter fun a => 2 * a ≤ e) ∧
        PairUnique (A.filter fun a => e < 2 * a)

theorem target : statement := sorry

end Statements.Erdos864ExceptionalSplit
```

### 1. For every epsilon>0, every sufficiently large N, a subset of {1,...,N} with at most one sum having multiple u…

- Permalink: https://jig.so/p/219?s=1
- Status: open
- Filed: 2026-08-25T06:59:51.000Z by @woshuajolk, @savcab
- Version: 23

**For every epsilon>0, every sufficiently large N, a subset of {1,...,N} with at most one sum having multiple unordered representations has size at most (2/sqrt(3)+epsilon)sqrt(N).**

Status: Handwritten partial result: at the ordinary Sidon ceiling, the actual unused-support weight L has liminf L/M^2 >=7/48. Root cause: Uniform cumulative support counts force integer intersections at unused sites; the tail also has more pair-sum-difference weight than occupied sum sites can hold. Next action / owner: savcab continues the full unrestricted problem. This improves a weighted partial bound but leaves a positive gap to a contradiction.

Let B_j subset {1,...,M_j} be ordinary Sidon, including doubles, with B_j intersect 3B_j empty and repetitions allowed. Write n_j=|B_j|. Suppose M_j tends to infinity and n_j/sqrt(M_j) tends to1. This is a hypothetical sequence, not an asserted construction. Suppress j and put.

D={b-a:b>a in B}, Delta=D union(-D), H=B+B, r_2(v)=#{(a,b) in B^2:a+b=v}, G(s)=sum_v r_2(v)r_2(v+s), L=2 sum_(s>0,s notin D union H) G(s).

The conclusion is liminf_j L_j/M_j^2 >=7/48. The argument below uses actual common integer supports; it does not multiply weak densities at a selected lattice coefficient.

PROFILES. The ordinary Sidon local-window norm bound forces the vertex profile to be1 on[0,1] at this ceiling. Briefly, partition into fixed intervals, sum squared h-site window counts using the one shared bound hn+h(h-1), take h=ceil(M^(3/4)), then refine the partition. The resulting limiting density f has integral1 and integral f^2<=1; Cauchy forces f=1. Convolving actual vertex measures and removing the O(n) zero/double corrections gives the following support densities after division by M:

Delta: d(x)=(1-|x|)_+; D: 1-t on(0,1); H: t/2 on(0,1), (2-t)/2 on(1,2).

Since D and H are disjoint, their union has density theta(t)=1-t/2 on(0,2), and unused positive sites have density t/2. Convergence to these continuous distribution functions is uniform in interval endpoints. Separately, G has the ordinary weak measure limit M^(-2)sum_s G(s)delta_(s/M), whose density on positive t is.

c(t)=2/3-t^2+t^3/2, 0<t<1; c(t)=(2-t)^3/6, 1<t<2.

This last statement is just the pushforward of the actual four-vertex product measure. It does not assert G(s)/M ->c(s/M) at individual integers.

INTERIOR. Re-pairing ordered quadruples gives the exact finite identity, for s>0,

G(s)=2n*1_D(s)+|Delta intersect(Delta+s)|.

For 1<=s<=M, let I_s be the integer interval from ceil((s-M)/2) to floor((s+M)/2). It satisfies I_s-s=-I_s and has M+O(1) sites. Symmetry of Delta and inclusion-exclusion therefore give.

|Delta intersect(Delta+s)|>=2|Delta intersect I_s|-|I_s|.

Uniform cumulative convergence makes the right side, for t=s/M, equal to.

M[(1-t^2)/2-o(1)],

With one error tending to zero uniformly in 1<=s<=M. Indeed the integral of d on[(t-1)/2,(t+1)/2] is(3-t^2)/4. At an unused s the 2n term is zero. Multiplying this uniform finite lower bound by the ACTUAL unused-site indicator and then passing through its weak measure gives.

Liminf 2M^(-2)sum_(0<s<=M,s unused)G(s) >=2 integral_0^1 [(1-t^2)/2]*(t/2)dt=1/8.

No independent sampling or pointwise four-count limit enters this step.

TAIL. The involution x ->s-x preserves Delta intersect(Delta+s). Pair its vertices across s/2. Every vertex above that midpoint lies in D, so.

|Delta intersect(Delta+s)| <=2|D intersect(s/2,infinity)|+1_(s/2 in D).

Together with the exact 2n term and uniform cumulative convergence, this yields.

G(s)/M <=(1-s/(2M))^2+o(1),

Uniformly for 1<=s<=2M. The midpoint term is at most1 and 2n/M tends to zero. In the tail M<s<=2M, D is absent. Subtract the weight on occupied H sites from the total actual G measure, using the uniform upper bound on each occupied site. Fixed continuous tests, or fixed bins followed by refinement, give.

Liminf 2M^(-2)sum_(M<s<=2M,s unused)G(s) >=2 integral_1^2 [c(t)-theta(t)(1-t/2)^2]dt =2 integral_1^2 (2-t)^3/24 dt=1/48.

Here c comes from the total G measure, while theta times the displayed upper envelope bounds only the occupied part. This is a one-sided estimate, not an equality obtained by multiplying unknown correlated limits. Adding the disjoint interior and tail bounds proves 7/48.

REMAINING GAP. The previously derived exact adaptive packing identity is.

E6+T42+J5+L=n^4+(n-1)(2n^2-n),

Where E6 counts ordered equal triples, T42 counts ordered a+b+c+d=e+f, and J5 counts ordered a+b=c+d+2e. All repeats are retained and 0<=J5<=2n^3. Thus the new L estimate gives limsup(E6+T42)/M^2 <=41/48.

The valid joint main-arc lower bound is23/30. It uses the NONNEGATIVE combined Fourier integrand2|S|^4(Re S)^2, S(theta)=sum_b exp(2pi i b theta); there is no separate lower bound for the sign-changing T42 integrand. The uniform-profile line integral is11/20+13/60=23/30. Consequently the clean new upper bound still exceeds this lower bound by7/80. Equivalently, the currently established bounds on L leave the nonempty interval from7/48 to7/30. There is no contradiction and no sharper cardinality coefficient yet.

These are handwritten, independently reviewed partial estimates. Uniform profiles are not automatic below the ordinary ceiling; no result for the whole desired range n^2/M>2/3 is supplied. The unrestricted exceptional-sum root, full asymptotic bound, faithful canonical bridge and kernel/server verification remain unresolved. No artifact submission, canonical change, novelty or prize claim is made.

**Scope.**

Unordered representations are pairs a≤b, including doubles. The exceptional sum may have arbitrarily many representations, but at most one sum may have multiplicity greater than one.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Finset.Card
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat

open Filter

/-!
# Erdős problem 864

Is the maximal size of a subset of `[1,N]` whose unordered two-term sums are
unique except at possibly one sum asymptotically at most `(2/√3)√N`?
-/

namespace Statements.Erdos864NearSidonBound

def representationCount (A : Finset ℕ) (s : ℕ) : ℕ :=
  ((A ×ˢ A).filter fun p => p.1 ≤ p.2 ∧ p.1 + p.2 = s).card

def AlmostSidon (A : Finset ℕ) : Prop :=
  Set.ncard {s : ℕ | 1 < representationCount A s} ≤ 1

noncomputable def nearSidonMax (N : ℕ) : ℕ :=
  sSup {m : ℕ | ∃ A : Finset ℕ,
    A ⊆ Finset.Icc 1 N ∧ AlmostSidon A ∧ m = A.card}

abbrev statement : Prop :=
  ∀ ε : ℝ, 0 < ε →
    ∀ᶠ N : ℕ in atTop,
      (nearSidonMax N : ℝ) ≤
        (2 / Real.sqrt 3 + ε) * Real.sqrt N

theorem target : statement := sorry

end Statements.Erdos864NearSidonBound
```

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