3.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

7 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 3

References:

erdosproblems.com/3
[BlSi20] Bloom, Thomas F. and Sisask, Olof, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).
[Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.
[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemerédi's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017).
[KeMe23] Kelley, Zander and Meka, Raghu, Strong bounds for 3-progressions. 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (2023).
[LSS24] Leng, James, Sah, Ashwin and Sawhney, Mehtaab, Improved bounds for Szemerédi's theorem. arXiv:2402.17995 (2024).

Imports (1)

FormalConjecturesUtil

theorem Erdos3.erdos_3 Erdos Problem 3: divergent reciprocal sums and arithmetic progressions sorry line 47 source
Source documentation
If $A \subset \mathbb{N}$ has $\sum_{n \in A}\frac 1 n = \infty$, then must $A$ contain arbitrarily long arithmetic progressions?
L4DEL explanation
If the reciprocals of a set of naturals sum to infinity, must the set contain arithmetic progressions of every length? The file marks this as open.
1. The attribute reads category research open, and the statement is written as "answer(sorry) iff ...", so the yes-or-no answer itself is left unfilled.
2. The right-hand side says: for every set A of naturals whose reciprocal sum is not summable, there are arbitrarily large k (infinitely often in k) with a subset S of A that is an arithmetic progression of length k.
3. The proof body is sorry, as for the other open problems in the repository.
Sources and attribution: Pinned Lean source (1646ca16afd6)
depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Not, Summable, Real, HDiv.hDiv, OfNat.ofNat, Membership.mem, SummationFilter.unconditional, Filter.Frequently, Exists, And, LE.le, Set.IsAPOfLength, ENat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,
    (¬ Summable fun a : A ↦ 1 / (a : ℝ)) →
    ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k := by
…
```
theorem Erdos3.erdos_3.variants.three The case of $3$-term progressions: if $A \subset \mathbb{N}$ has $\sum_{n \in A}\frac 1 n = \infty$, then $A$ contains a non-trivial $3$-term arithmetic progression sorry line 59 source
Source documentation
The case of $3$-term progressions: if $A \subset \mathbb{N}$ has $\sum_{n \in A}\frac 1 n = \infty$, then $A$ contains a non-trivial $3$-term arithmetic progression.
Proved by Bloom and Sisask [BlSi20].

depends on sorry what this means
Statement uses: Set, Nat, Not, Summable, Real, HDiv.hDiv, OfNat.ofNat, Membership.mem, SummationFilter.unconditional, Exists, And, LE.le, Set.IsAPOfLength, ENat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_3.variants.three : ∀ A : Set ℕ,
    (¬ Summable fun a : A ↦ 1 / (a : ℝ)) → ∃ S ⊆ A, S.IsAPOfLength 3 := by
…
```
abbrev Erdos3.r $r_k(N)$ is the largest size of a subset of $\{1, \dots, N\}$ that does not contain a non-trivial $k$-term arithmetic progression line 67 source
Source documentation
$r_k(N)$ is the largest size of a subset of $\{1, \dots, N\}$ that does not contain a non-trivial $k$-term arithmetic progression.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.IsAPOfLengthFree.maxCard
```
noncomputable abbrev r := Set.IsAPOfLengthFree.maxCard
…
```
Mentions: Set.IsAPOfLengthFree.maxCard
theorem Erdos3.erdos_3.variants.kelley_meka There are $\beta > 0$ and $c > 0$ such that $r_3(N) \ll N \exp(-c (\log N)^{\beta})$ sorry line 75 source
Source documentation
There are $\beta > 0$ and $c > 0$ such that $r_3(N) \ll N \exp(-c (\log N)^{\beta})$.
Proved by Kelley and Meka [KeMe23].

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Erdos3.r, HMul.hMul, Real.exp, Neg.neg, HPow.hPow, Real.log
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_3.variants.kelley_meka : ∃ β > (0 : ℝ), ∃ c > (0 : ℝ),
    (fun N ↦ (r 3 N : ℝ)) ≪ fun N : ℕ ↦ (N : ℝ) * Real.exp (-c * Real.log N ^ β) := by
…
```
Mentions: c, Erdos3.r
theorem Erdos3.erdos_3.variants.green_tao There is $c > 0$ such that $r_4(N) \ll N (\log N)^{-c}$ sorry line 85 source
Source documentation
There is $c > 0$ such that $r_4(N) \ll N (\log N)^{-c}$.
Proved by Green and Tao [GrTa17].

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Erdos3.r, HMul.hMul, HPow.hPow, Real.log, Neg.neg
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_3.variants.green_tao : ∃ c > (0 : ℝ),
    (fun N ↦ (r 4 N : ℝ)) ≪ fun N : ℕ ↦ (N : ℝ) * Real.log N ^ (-c) := by
…
```
Mentions: c, Erdos3.r
theorem Erdos3.erdos_3.variants.gowers For every $k \geq 3$ there is $c_k > 0$ such that $r_k(N) \ll_k N (\log \log N)^{-c_k}$ sorry line 95 source
Source documentation
For every $k \geq 3$ there is $c_k > 0$ such that $r_k(N) \ll_k N (\log \log N)^{-c_k}$.
Proved by Gowers [Go01].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, GT.gt, Asymptotics.IsBigO, Filter.atTop, Erdos3.r, HMul.hMul, HPow.hPow, Real.log, Neg.neg
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_3.variants.gowers (k : ℕ) (hk : 3 ≤ k) : ∃ c > (0 : ℝ),
    (fun N ↦ (r k N : ℝ)) ≪ fun N : ℕ ↦ (N : ℝ) * Real.log (Real.log N) ^ (-c) := by
…
```
Mentions: c, Erdos3.r
theorem Erdos3.erdos_3.variants.leng_sah_sawhney For every $k \geq 5$ there is $c_k > 0$ such that $r_k(N) \ll_k N \exp(-(\log \log N)^{c_k})$ sorry line 105 source
Source documentation
For every $k \geq 5$ there is $c_k > 0$ such that $r_k(N) \ll_k N \exp(-(\log \log N)^{c_k})$.
Proved by Leng, Sah and Sawhney [LSS24].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, GT.gt, Asymptotics.IsBigO, Filter.atTop, Erdos3.r, HMul.hMul, Real.exp, Neg.neg, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_3.variants.leng_sah_sawhney (k : ℕ) (hk : 5 ≤ k) : ∃ c > (0 : ℝ),
    (fun N ↦ (r k N : ℝ)) ≪
      fun N : ℕ ↦ (N : ℝ) * Real.exp (-Real.log (Real.log N) ^ c) := by
…
```
Mentions: c, Erdos3.r