987.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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

11 declarations, 10 contain sorry, 0 declared as axioms

Erdős Problem 987

References:

erdosproblems.com/987
[APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics, probability, and number theory II. arXiv:2604.06609 (2026).
[Cl67] Clunie, J., On a problem of Erdős. J. London Math. Soc. (1967), 133--136.
[Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.
[Er65b] Erdős, P., Some remarks on number theory. Israel J. Math. (the actual reference cited by Clunie 1967 as [2]; the erdosproblems.com bibliography points to a different Erdős 1965 paper, "Some recent advances and current problems in number theory" (Lectures on Modern Mathematics III, 1965, 196-244), which does not appear to contain the exponential-sum log-bound proof).
[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.
[Li69] Lindström, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.

Imports (1)

FormalConjecturesUtil

def Erdos987.A For an infinite sequence $x_1, x_2, \ldots \in (0, 1)$, define $$A_k = \limsup_{n \to \infty} \left\lvert \sum_{j \le n} e(k x_j) \right\rvert,$$ where $e(x) = e^{2\pi i x}$ line 58 source
Source documentation
For an infinite sequence $x_1, x_2, \ldots \in (0, 1)$, define $$A_k = \limsup_{n \to \infty} \left\lvert \sum_{j \le n} e(k x_j) \right\rvert,$$ where $e(x) = e^{2\pi i x}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, EReal
Proof uses: Filter.limsup, Norm.norm, Complex, Finset.sum, Finset.range, additiveChar, HMul.hMul, Filter.atTop
12 instances, coercions and auxiliary constants left out.
```
noncomputable def A (x : ℕ → ℝ) (k : ℕ) : EReal :=
…
```
theorem Erdos987.erdos_987.parts.i sorry line 73 source
depends on sorry what this means
Statement uses: Iff, True, Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Eq, EReal, Filter.limsup, Erdos987.A, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.sup_limsup_infty sorry line 83 source
depends on sorry what this means
Statement uses: Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Eq, EReal, Filter.limsup, iSup, Norm.norm, Complex, Finset.sum, Finset.range, additiveChar, HMul.hMul, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
41 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.log_lower_bound sorry line 93 source
depends on sorry what this means
Statement uses: Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Exists, And, GT.gt, Filter.Frequently, LE.le, EReal, HMul.hMul, Real.log, Erdos987.A, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.sqrt_lower_bound sorry line 102 source
depends on sorry what this means
Statement uses: Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Exists, And, GT.gt, Filter.Frequently, LE.le, EReal, HMul.hMul, Real.sqrt, Erdos987.A, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.linear_upper_bound_weak sorry line 113 source
depends on sorry what this means
Statement uses: Exists, Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, LE.le, EReal, Erdos987.A, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.linear_upper_bound_clunie sorry line 125 source
depends on sorry what this means
Statement uses: Exists, Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, LE.le, EReal, Erdos987.A, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.linear_upper_bound sorry line 141 source
depends on sorry what this means
Statement uses: Exists, Nat, Real, Membership.mem, Set, Set.Ico, OfNat.ofNat, LE.le, EReal, Erdos987.A
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 Erdos987.erdos_987.variants.sqrt_log_upper_bound sorry line 155 source
depends on sorry what this means
Statement uses: Exists, Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, LT.lt, LE.le, Norm.norm, Complex, Finset.sum, Finset.range, additiveChar, HMul.hMul, Real.sqrt, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.parts.ii sorry line 168 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, And, Asymptotics.IsLittleO, Filter.atTop, Filter.Eventually, LE.le, EReal, Erdos987.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos987.erdos_987.variants.finite_distinct_points sorry line 179 source
depends on sorry what this means
Statement uses: Nat, Real, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Set.Finite, Set.range, Filter.Frequently, Eq, EReal, Erdos987.A, Top.top, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.