67.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 67

References:

erdosproblems.com/67
[Ta16] Tao, Terence, The Erdős discrepancy problem. Discrete Anal. (2016), Paper No. 1, 29.

Imports (1)

FormalConjecturesUtil

theorem Erdos67.erdos_67 Documented declaration: erdos_67 sorry line 42 source
Source documentation
The Erdős discrepancy problem
If $f\colon \mathbb N \rightarrow \{-1, +1\}$ then is it true that for every $C>0$ there exist $d, m \ge 1$ such that $$\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?$$ This is true, and was proved by Tao [Ta16]

depends on sorry what this means
Statement uses: Nat, Subtype, Real, Membership.mem, Finset, Insert.insert, Neg.neg, OfNat.ofNat, Singleton.singleton, LT.lt, Exists, And, GE.ge, abs, Finset.sum, Finset.Icc, HMul.hMul
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_67 (f : ℕ → ({-1, 1} : Finset ℝ)) (C : ℝ) (hC : 0 < C) : ∃ᵉ (d ≥ 1) (m ≥ 1),
    C < |∑ k ∈ Finset.Icc 1 m, (f (k * d)).1| := by
…
```
theorem Erdos67.erdos_67.variants.complex Documented declaration: complex sorry line 54 source
Source documentation
The Erdős discrepancy problem (complex variant)
If $f\colon \mathbb N \rightarrow S^1 ⊆ ℂ$ then is it true that for every $C>0$ there exist $d, m \ge 1$ such that $$\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?$$ This is true, and was proved by Tao [Ta16]

depends on sorry what this means
Statement uses: Nat, Complex, Metric.sphere, OfNat.ofNat, Real, LT.lt, Exists, And, GE.ge, Norm.norm, Finset.sum, Finset.Icc, Membership.mem, Set, HMul.hMul
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_67.variants.complex (f : ℕ → Metric.sphere (0 : ℂ) 1) (C : ℝ) (hC : 0 < C) :
    ∃ᵉ (d ≥ 1) (m ≥ 1), C < ‖∑ k ∈ Finset.Icc 1 m, (f (k * d)).1‖  := by
…
```