Add Erdos 5

FormalConjectures/ErdosProblems/5.lean

@@ -0,0 +1,92 @@
+/-
+Copyright 2025 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
+
+/-!
+# Erdős Problem 5
+
+*Reference:* [erdosproblems.com/5](https://www.erdosproblems.com/5)
+-/
+
+open Real Filter Function Set
+open scoped Topology
+
+namespace erdos_5
+
+/--
+There exists a strictly increasing sequence of indices (n_i)
+such that the limit of $(p_{n_{i+1}} - p_{n_i}) / log p_{n_i}$ equals $\infty$,
+where $p_k$ denotes the k-th prime number.
+
+**Note:**
+This definition is equivalent to $(p_{n_{i+1}} - p_{n_i}) / log n_i$
+Since $p_n \sim n \log n$, then $\log n \sim \log p_n$
+-/
+@[category research solved, AMS 11]
+theorem infinite_case :
+    ∃ p : ℕ → ℕ, StrictMono p ∧ (∀ n : ℕ, (p n).Prime) ∧
+    Tendsto (fun i ↦ (p (i+1) - p i) / log (p i)) atTop atTop := by
+  sorry
+
+/--
+**The set of limit points of normalized prime gaps**
+-/
+def limit_points : Set ℝ :=
+  {x : ℝ | ∃ p : ℕ → ℕ, StrictMono p ∧ (∀ n : ℕ, (p n).Prime) ∧
+    Tendsto (fun i ↦ (p (i+1)  - p i) / log (p i)) atTop (𝓝 x) }
+
+/--
+For every nonnegative real number x, there exists a strictly increasing sequence of indices (n_i)
+such that the limit of $(p_{n_{i+1}} - p_{n_i}) / log p_{n_i}$ equals $x$,
+where $p_k$ denotes the k-th prime number.
+-/
+@[category research open, AMS 11]
+theorem finite_case :
+    {x : ℝ | 0 ≤ x} = limit_points := by
+  sorry
+
+/--
+`limit_points` is everywhere dense.
+-/
+@[category research open, AMS 11]
+theorem everywhere_dense :
+    {x : ℝ | 0 ≤ x} = closure limit_points := by
+  sorry
+
+/--
+At least 1/3 of positive real numbers are in the set of `limit_points` ($S$).
+More precisely, the Lebesque measure of $S ∩ [0, c]$ is at least $c/3$.
+
+[Me20] Merikoski, J. (2020) Limit points of normalized prime gaps.
+-/
+@[category research solved, AMS 11]
+theorem contains_at_least_1_3 (c : ℝ) (hc : 0 ≤ c) :
+    MeasureTheory.volume (limit_points ∩ Icc 0 c) > ENNReal.ofReal (c / 3) := by
+  sorry
+
+/--
+There exists some constant $c$ such that $[0,c] \subset S$, where $S$ is `limit_points`.
+
+[Pi16] Pintz, J. (2016) Polignac Numbers, Conjectures of Erdős on Gaps between
+Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture.
+-/
+@[category research solved, AMS 11]
+theorem contains_interval :
+    ∃ (c : ℝ) (hc : 0 < c), {x : ℝ | 0 ≤ x ∧ x ≤ c} ⊆ limit_points := by
+  sorry
+
+end erdos_5