1206.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1206

References:

erdosproblems.com/1206
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[GGK26] M. Garaev, F. Garayev, and S. Konyagin, On Sidon sets with squares, cubes, and quartics in short intervals. arXiv:2602.08807 (2026).
[GaKo24] Gabdullin, M. R. and Konyagin, S. V., Trigonometric polynomials with frequencies in the set of cubes. Math. Notes (2024), 336--340.

Imports (1)

FormalConjecturesUtil

theorem Erdos1206.erdos_1206.parts.i Does $\{1,2^3,\ldots,N^3\}$ contain a Sidon set of size $\gg N$? sorry line 41 source
Source documentation
Does $\{1,2^3,\ldots,N^3\}$ contain a Sidon set of size $\gg N$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Finset, LE.le, Finset.image, HPow.hPow, Finset.Icc, IsSidon, HMul.hMul, Finset.card, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1206.parts.i : answer(sorry) ↔
    ∃ c : ℝ, 0 < c ∧ ∀ᶠ N in Filter.atTop, ∃ S : Finset ℕ,
      S ⊆ (Finset.Icc 1 N).image (fun n => n ^ 3) ∧
      IsSidon (S : Set ℕ) ∧ c * (N : ℝ) ≤ (S.card : ℝ) := by
…
```
Mentions: c, IsSidon
theorem Erdos1206.erdos_1206.parts.ii Is there an infinite set $A\subset \mathbb{N}$ of positive density such that $\{a^3 : a\in A\}$ is a Sidon set? sorry line 51 source
Source documentation
Is there an infinite set $A\subset \mathbb{N}$ of positive density such that $\{a^3 : a\in A\}$ is a Sidon set?

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Set.Infinite, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Set.univ, IsSidon, Set.image, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1206.parts.ii : answer(sorry) ↔
    ∃ A : Set ℕ, A.Infinite ∧ 0 < A.lowerDensity ∧
      IsSidon ((fun a : ℕ => a ^ 3) '' A) := by
…
```
Mentions: IsSidon