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import FormalConjecturesUtilErdős Problem 296
[Bl21] Bloom, T. F.,
namespace Erdos296
open FilterThe reciprocal sum of a finite set of natural numbers, as a rational number.
def recipSum (A : Finset ℕ) : ℚ :=
∑ n ∈ A, (1 : ℚ) / n
There are k pairwise disjoint subsets of {1, ..., N} with reciprocal sum 1.
def HasDisjointUnitDecomps (N k : ℕ) : Prop :=
∃ f : Fin k → Finset ℕ,
(∀ i, f i ⊆ Finset.Icc 1 N) ∧
(∀ i, recipSum (f i) = 1) ∧
(∀ i j : Fin k, i ≠ j → Disjoint (f i) (f j))
Let $N\geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,\ldots,A_k\subseteq {1,\ldots,N}$ with $\sum_{n\in A_i}\frac{1}{n}=1$ for all $i$. Estimate $k(N)$. Is it true that $k(N)=o(\log N)$?
Hunter and Sawhney observed that Bloom's theorem [Bl21], together with the greedy argument, gives $k(N)=(1-o(1))\log N$.
@[category research solved, AMS 11,
formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos296.lean"]
theorem erdos_296 :
(∀ N k : ℕ, HasDisjointUnitDecomps N k → (k : ℚ) ≤ recipSum (Finset.Icc 1 N)) ∧
(∀ ε : ℝ, 0 < ε → ε < 1 → ∀ᶠ N : ℕ in atTop,
HasDisjointUnitDecomps N ⌊(1 - ε) * Real.log N⌋₊) := ⊢ (∀ (N k : ℕ), HasDisjointUnitDecomps N k → ↑k ≤ recipSum (Finset.Icc 1 N)) ∧
∀ (ε : ℝ), 0 < ε → ε < 1 → ∀ᶠ (N : ℕ) in atTop, HasDisjointUnitDecomps N ⌊(1 - ε) * Real.log ↑N⌋₊
All goals completed! 🐙
end Erdos296