Erdős Problem 296

References:

• erdosproblems.com/296
• [Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).

def Erdos296.recipSum (A : Finset ℕ) : ℚ

The reciprocal sum of a finite set of natural numbers, as a rational number.

Equations

• Erdos296.recipSum A = ∑ n ∈ A, 1 / ↑n

def Erdos296.HasDisjointUnitDecomps (N k : ℕ) : Prop

There are k pairwise disjoint subsets of {1, ..., N} with reciprocal sum 1.

Equations

• One or more equations did not get rendered due to their size.

theorem Erdos296.erdos_296 : (∀ (N k : ℕ), HasDisjointUnitDecomps N k → ↑k ≤ recipSum (Finset.Icc 1 N)) ∧ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∀ᶠ (N : ℕ) in Filter.atTop, HasDisjointUnitDecomps N ⌊(1 - ε) * Real.log ↑N⌋₊

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 $k(N)=(1-o(1))\log N$.