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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+
+/-!
+# Ben Green's Open Problem 82
+
+Let $A \subset \mathbb{Z}$ be a set of size $n$. For how many $\theta \in \mathbb{R}/\mathbb{Z}$
+must we have $\sum_{a \in A} \cos(2\pi a\theta) = 0$?
+
+It is known that there must be at least $\Omega(\log n / \log \log n)$ zeros.
+
+*Reference:*
+- [Gr24] [Ben Green's Open Problem 82](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.82)
+-/
+
+open Real Set
+open scoped Finset
+
+namespace Green82
+
+/--
+Let $A \subset \mathbb{Z}$ be a set of size $n$. For how many $\theta \in \mathbb{R}/\mathbb{Z}$
+must we have $\sum_{a \in A} \cos(2\pi a\theta) = 0$?
+-/
+@[category research open, AMS 11 42]
+theorem green_82 :
+    answer(sorry) ↔ ∃ f : ℕ → ℕ, ∀ n > 0, ∀ A : Finset ℤ, A.card = n →
+      f n ≤ ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard := by
+  sorry
+
+/-- There must be at least $\Omega(\log n / \log \log n)$ zeros. -/
+@[category research solved, AMS 11 42]
+theorem green_82_lower_bound :
+    ∃ c > 0, ∀ n ≥ 2, ∀ A : Finset ℤ, A.card = n →
+      c * log n / log (log n) ≤
+        ({θ : ℝ | θ ∈ Ico 0 1 ∧ ∑ a ∈ A, cos (2 * π * a * θ) = 0} : Set ℝ).ncard := by
+  sorry
+
+end Green82