Feat: formalise Erdős Problem 359

FormalConjectures/ErdosProblems/359.lean

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+/-
+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 359
+
+*Reference:* [erdosproblems.com/359](https://www.erdosproblems.com/359)
+-/
+
+namespace Erdos359
+
+open Filter Asymptotics
+
+/-- The predicate that `A` is monotone, `A 0 = n` and for all `j`, `A (j + 1)` is the smallest natural number that
+cannot be written as a sum of consecutive terms of `A 0, ..., A j` -/
+def IsGoodFor (A : ℕ → ℕ) (n : ℕ) : Prop := A 0 = n ∧ StrictMono A ∧
+  ∀ j, IsLeast
+    {m : ℕ | A j < m ∧ ∀ a b, Finset.Icc a b ⊆ Finset.Iic j → m ≠ ∑ i ∈ Finset.Icc a b, A i}
+    (A <| j + 1)
+
+/-- Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the
+least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \to \infty$. -/
+@[category research open, AMS 11]
+theorem erdos_359.parts.i (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
+    atTop.Tendsto (fun k ↦ (A k : ℝ) / k) atTop := by
+  sorry
+
+/-- Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the
+least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \to 0$
+for any $c > 0$. -/
+@[category research open, AMS 11]
+theorem erdos_359.parts.ii (A : ℕ → ℕ) (hA : IsGoodFor A 1) (c : ℝ) (hc : 0 < c):
+    atTop.Tendsto (fun k ↦ A k / (k : ℝ) ^ (1 + c)) atTop := by
+  sorry
+
+/-- Suppose monotone sequence $A$ satisfies the following: `A 0 = 1` and for all `j`, `A (j + 1)` is the
+smallest natural number that cannot be written as a sum of consecutive terms of `A 0, ..., A j`.
+Then the first few terms of $A$ are $1,2,4,5,8,10,14,15,...$. -/
+@[category test, AMS 11]
+theorem erdos_359.variants.isGoodFor_1_low_values (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
+    A '' (Set.Iic 7) = {1, 2, 4, 5, 8, 10, 14, 15} := by
+  sorry
+
+/-- Suppose monotone sequence $A$ satisfies the following: `A 0 = 1` and for all `j`, `A (j + 1)` is the
+smallest natural number that cannot be written as a sum of consecutive terms of `A 0, ..., A j`.
+Then it is conjectured that $$a_k ~ \frac{k \log k}{\log \log k}$$. -/
+@[category research open, AMS 11]
+theorem erdos_359.variants.isGoodFor_1_asymptotic (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
+    (fun k ↦ (A k : ℝ)) ~[atTop] (fun k ↦ k * (k : ℝ).log / (k : ℝ).log.log) := by
+  sorry
+
+end Erdos359