erdos_67.variants.complex

**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]

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‖

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