FormalConjectures.ErdosProblems.«67»

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import FormalConjecturesUtil

Erdős Problem 67

References:

erdosproblems.com/67

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

open Filter


namespace Erdos67

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]

@[category research solved, AMS 11]
theorem declaration uses 'sorry'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| := byf:ℕ → ↥{-1, 1}C:ℝhC:0 < C⊢ ∃ d ≥ 1, ∃ m ≥ 1, C < |∑ k ∈ Finset.Icc 1 m, ↑(f (k * d))|
  sorryAll goals completed! 🐙

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]

@[category research solved, AMS 11]
theorem declaration uses 'sorry'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‖  := byf:ℕ → ↑(Metric.sphere 0 1)C:ℝhC:0 < C⊢ ∃ d ≥ 1, ∃ m ≥ 1, C < ‖∑ k ∈ Finset.Icc 1 m, ↑(f (k * d))‖
  sorryAll goals completed! 🐙



end Erdos67