feat: add Green's Open Problem 82 #1897
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Signed-off-by: krrish175-byte <krrishbiswas175@gmail.com>
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| $A \subset \mathbb{Z}$ with $|A| = n$. | ||
| -/ | ||
| @[category research open, AMS 11 42] | ||
| theorem green_82 (n : ℕ) (hn : 1 ≤ n) : |
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n probably shouldnt be in the context as usual
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| ⨅ (A : Finset ℤ) (_ : A.card = n), | ||
| ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard = |
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Better write this as an inequality and not take sup?
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The green problem list mentions known bounds, please include them. |
| @[category research open, AMS 11 42] | ||
| theorem green_82 (n : ℕ) (hn : 1 ≤ n) : | ||
| ⨅ (A : Finset ℤ) (_ : A.card = n), | ||
| ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard = |
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This is not what the problem is asking. Why do you have to limit Ico 0 1?
Signed-off-by: krrish175-byte <krrishbiswas175@gmail.com>
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Thanks for the review. Sorry, I missed those. I have updated the code as you asked for:
Regarding Ico 0 1: Since the trigonometric polynomial is 1-periodic, counting zeros in the fundamental domain [0, 1) seemed like the most direct way to represent the count on R / Z. Let me know if you would prefer using UnitAddCircle instead. Thanks for the review again! |
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Closes #1737
This PR formalizes Green's Open Problem 82, which asks: for a finite set A ⊂ ℤ of size n, what is the minimum number of zeros that ∑_{a∈A} cos(2πaθ) must have on ℝ/ℤ?