FormalConjectures.ErdosProblems.«845»

/-
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 FormalConjecturesUtil

Erdős Problem 845

Reference: erdosproblems.com/845


namespace Erdos845

Let $C > 0$. Is it true that the set of integers of the form $n = b_1 + \cdots + b_t$, with $b_1 < \cdots < b_t$, where $b_i = 2^{k_i}3^{l_i}$ for $1 \leq i\leq t$ and $b_t \leq Cb_1$ has density $0$?

van Doorn and Everts \cite{vDEv25} have disproved this with $C=6$ - in fact, they prove that all integers can be written as such a sum in which $b_t<6b_1$.

This was formalized in Lean by Alexeev using Aristotle.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos845.lean"]
theorem declaration uses 'sorry'erdos_845 :
    answer(False) ↔
      ∀ᵉ (C : ℝ) (hC : 0 < C),
        let f : ℕ × ℕ → ℕ := fun (k, l) ↦ 2 ^ k * 3 ^ l
        { ∑ x ∈ B, f x | (B : Finset (ℕ × ℕ)) (h : B.Nonempty)
          (hB : B.sup f ≤ C * B.inf' h f) }.HasDensity 0 := by⊢ False ↔
  ∀ (C : ℝ),
    0 < C →
      let f := fun x =>
        match x with
        | (k, l) => 2 ^ k * 3 ^ l;
      {x | ∃ B, ∃ (h : B.Nonempty) (_ : ↑(B.sup f) ≤ C * ↑(B.inf' h f)), ∑ x ∈ B, f x = x}.HasDensity 0
  sorryAll goals completed! 🐙


end Erdos845