feat(GreensOpenProblems): 39 (#3804)

# Description
> If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can
we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$
translates of $A$?

- Closes #1595
- 🤖 First draft created with help from **Gemini 3.1 Pro**. I then
modified it to handle edge cases, improved code modularity, added
variants, and simple tests. I then asked Gemini to come up with
non-trivial tests by enumeration.
- Formulation handles the 3 problem variants mentioned in Green's
original comments.

# Testing
✅ Builds successfully
```shell
$ lake build FormalConjectures/GreensOpenProblems/39.lean 
Build completed successfully (7982 jobs).
```

Author: jeangud

FormalConjectures/GreensOpenProblems/39.lean

@@ -0,0 +1,119 @@
+/-
+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
+
+/-!
+# Green's Open Problem 39
+
+*References:*
+- [Gr24] [Green, Ben. "100 open problems." (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.39)
+- [BJR11] Bollobás, Béla, Svante Janson, and Oliver Riordan. "On covering by translates of a set."
+  Random Structures & Algorithms 38.1‐2 (2011): 33-67.
+-/
+
+open Filter Topology
+open scoped Pointwise
+
+namespace Green39
+
+/--
+The proportion of subsets of $\mathbb{Z}/p\mathbb{Z}$ of size $k$ that can cover
+$\mathbb{Z}/p\mathbb{Z}$ using at most $c$ translates.
+
+If p = 0 or k > p, return 0 by convention.
+-/
+def proportionCoverable (p k c : ℕ) : ℚ :=
+  if h : p = 0 then 0
+  else if k > p then 0
+  else
+    have : NeZero p := ⟨h⟩
+    let S : Finset (Finset (ZMod p)) := Finset.powersetCard k Finset.univ
+    let coverable := S.filter (fun A => ∃ T : Finset (ZMod p), T.card ≤ c ∧ A + T = Finset.univ)
+    (coverable.card : ℚ) / (S.card : ℚ)
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_p_p_1 : proportionCoverable 3 3 1 = 1 := by native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_t_0 : proportionCoverable 5 2 0 = 0 := by native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_2_1_2 : proportionCoverable 2 1 2 = 1 := by native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_3_1_2 : proportionCoverable 3 1 2 = 0 := by native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_a_gt_p : proportionCoverable 3 4 2 = 0 := by native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_7_4_2 :
+    proportionCoverable 7 4 2 = (3 : ℚ) / 5 := by
+  native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_11_3_4 :
+    proportionCoverable 11 3 4 = (1 : ℚ) / 3 := by
+  native_decide
+
+@[category test, AMS 5 60]
+theorem proportionCoverable_11_4_3 :
+    proportionCoverable 11 4 3 = (1 : ℚ) / 6 := by
+  native_decide
+
+/--
+If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can we almost surely cover
+$\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$? [Gr24]
+-/
+@[category research open, AMS 5 60]
+theorem green_39 : answer(sorry) ↔
+    Tendsto
+      (fun p : {q : ℕ // q.Prime} ↦
+        let k := Nat.sqrt p
+        let c := 100 * k
+        (proportionCoverable p k c : ℝ))
+      atTop (𝓝 1) := by
+  sorry
+
+/--
+"I do not know how to answer this even with 100 replaced by 1.01." [Gr24]"
+-/
+@[category research open, AMS 5 60]
+theorem green_39.variant_101 : answer(sorry) ↔
+    Tendsto
+      (fun p : {q : ℕ // q.Prime} ↦
+        let k := Nat.sqrt p
+        let c := ⌊1.01 * (k : ℝ)⌋₊
+        (proportionCoverable p k c : ℝ))
+      atTop (𝓝 1) := by
+  sorry
+
+/--
+Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$. [Gr24]
+-/
+@[category research open, AMS 5 60]
+theorem green_39.variant_theta : answer(sorry) ↔
+    ∀ (θ : ℝ), 0 < θ → θ ≤ 1/2 →
+    ∃ C > 1, Tendsto
+      (fun p : {q : ℕ // q.Prime} ↦
+        let k := ⌊(p : ℝ) ^ θ⌋₊
+        let c := ⌊C * (p : ℝ) ^ θ⌋₊
+        (proportionCoverable p k c : ℝ))
+      atTop (𝓝 1) := by
+  sorry
+
+end Green39