feat(GreensOpenProblems): 2

FormalConjectures/GreensOpenProblems/2.lean

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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
+
+open Filter
+open scoped Topology
+
+/-!
+# Ben Green's Open Problem 2
+
+References:
+- [Gr24] [Green, Ben. "100 open problems." (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.2)
+- [Er65] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII,
+  pages 181–189. Amer. Math. Soc., Providence, R.I., 1965.
+- [Sa21] Sanders, Tom. "The Erdős–Moser Sum-free Set Problem." Canadian Journal of Mathematics 73.1
+  (2021): 63-107.
+- [Ru05] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.
+- [Ch71] S. L. G. Choi. On a combinatorial problem in number theory. Proc. London Math. Soc. (3),
+  23:629–642, 1971. doi:10.1112/plms/s3-23.4.629.
+- [BSS00] A. Baltz, T. Schoen, and A. Srivastav. Probabilistic construction of small strongly
+  sum-free sets via large Sidon sets. Colloq. Math., 86(2):171–176, 2000.
+  doi:10.4064/cm-86-2-171-176.
+-/
+namespace Green2
+
+/--
+The restricted sumset of a set $S$, denoted $S \hat{+} S$, is the set
+$\lbrace s_1 + s_2 : s_1, s_2 \in S, s_1 \neq s_2 \rbrace$.
+-/
+def restrictedSumset (S : Finset ℤ) : Finset ℤ :=
+  S.offDiag.image (fun p => p.1 + p.2)
+
+/--
+We define the construction from [Sa21, p1] as
+$M(A) := \max \{|S| : S \subseteq A \text{ and } (S \hat{+} S) \cap A = \varnothing \}$.
+-/
+def maxRestrictedSumAvoidingSubsetSize (A : Finset ℤ) : ℕ :=
+  (A.powerset.filter fun S => Disjoint (restrictedSumset S) A).sup Finset.card
+
+/--
+Let $A \subset \mathbf{Z}$ be a set of $n$ integers. Is there a set $S \subset A$ of size
+$(\log n)^{100}$ such that the restricted sumset$S \hat{+} S$ is disjoint from $A$?
+-/
+@[category research open, AMS 11]
+theorem green_2 : answer(sorry) ↔
+    ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
+      (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≥ (Real.log n) ^ 100 := by
+  sorry
+
+/--
+From [Sa21] it is known that there is always such an S with $|S| \gt (\log |A|)^{1+c}$.
+-/
+@[category research solved, AMS 11]
+theorem green_2_lower_bound_sanders :
+    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
+      maxRestrictedSumAvoidingSubsetSize A ≥ Real.log n ^ (1 + c) := by
+  sorry
+
+/--
+From [Er65] it is known that $M(A) \le \frac{1}{3}|A| + O(1)$.
+-/
+@[category research solved, AMS 11]
+theorem green_2_upper_bound_erdos :
+    ∃ C : ℝ, ∀ A : Finset ℤ,
+      (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≤ A.card / 3 + C := by
+  sorry
+
+/-
+TODO(jeangud): Add additional bounds on non-negative integers from Selfridge [Er65, p187], and
+Choi [Er65, p190], and [BSS00].
+-/
+
+/--
+From [Ch71] it is known that $M(A) \le |A|^{2/5 + o(1)}$.
+-/
+@[category research solved, AMS 11]
+theorem green_2_upper_bound_choi :
+    ∃ (o : ℕ → ℝ) (_ : Tendsto o atTop (𝓝 0)), ∀ A : Finset ℤ,
+      (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≤ A.card ^ (2 / 5 + o A.card) := by
+  sorry
+
+/--
+From [Ru05] the best-known upper bound is $|S| \lt e^{C \sqrt{\log |A|}}$.
+-/
+@[category research solved, AMS 11]
+theorem green_2_upper_bound_ruzsa :
+    ∃ C > (0 : ℝ), ∀ A : Finset ℤ,
+      (maxRestrictedSumAvoidingSubsetSize A : ℝ) < Real.exp (C * Real.sqrt (Real.log A.card)) := by
+  sorry
+
+end Green2