For a finite set S⊂N, define its Egyptian sum by E(S)=∑_m∈ S1/m. It is an underapproximation of x∈R if every m∈ S is positive and E(S) 0 and there exists a strictly increasing sequence (m_k)_k∈N of positive natural numbers and an n_0∈N such that, for every n≥ n_0, the set m_0,…,m_n-1 is a best n-term underapproximation of x. Is it true that for almost every x∈(0,∞), with respect to Lebesgue measure, x is eventually greedy?

Erdős Problem #206 — Eventually greedy Egyptian-fraction underapproximations

At least 45 years old · documented by 1980

For a finite set $S\subset\mathbb{N}$ , define its Egyptian sum by $E(S)=\sum_{m\in S}1/m$ . It is an underapproximation of $x\in\mathbb{R}$ if every $m\in S$ is positive and $E(S)<x$ . It is a best $n$ -term underapproximation of $x$ if $|S|=n$ , it is an underapproximation of $x$ , and every $n$ -element underapproximation $T$ of $x$ satisfies $E(T)\le E(S)$ . Say that $x$ is eventually greedy if $x>0$ and there exists a strictly increasing sequence $(m_k)_{k\in\mathbb{N}}$ of positive natural numbers and an $n_0\in\mathbb{N}$ such that, for every $n\ge n_0$ , the set $\{m_0,\ldots,m_{n-1}\}$ is a best $n$ -term underapproximation of $x$ . Is it true that for almost every $x\in(0,\infty)$ , with respect to Lebesgue measure, $x$ is eventually greedy?

References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

number-theory unit-fractions

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 paper shows that, for almost every real number, the greedy method eventually fails to give the best Egyptian-fraction approximations.

The problem asks whether almost every positive real $x$ eventually has its best $n$ -term unit-fraction underapproximations generated greedily. Erdős and Graham raised the question in 1980, suggesting an affirmative answer; it became Erdős Problem $206$ .

Known results

Curtiss, Takenouchi, and Soundararajan: equality for $x=1$ .
Erdős: equality for every unit fraction $x=1/b$ .
Nathanson: equality for rationals $x=a/b$ with $a$ dividing $b+1$ .
Chu: further rational cases, including odd $b$ with the stated divisibility condition.

July 2026 measure-zero theorem

Theorem 1 of the 2026 paper proves that the set of positive reals with the eventual-greedy property has Lebesgue measure zero, decisively answering Problem $206$ negatively. The proof finds a positive proportion of non-greedy best two-term approximations in infinitely many intervals. A corollary gives a transcendental counterexample, non-constructively. The paper leaves the corresponding question for positive rationals open.

Current status (as of July 2026): The almost-everywhere question is resolved negatively; the rational-number case remains open.

Sources

No solutions have been posted yet.