Erdős Problem #311

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Only nonzero approximation errors enter the minimum; exact representations of one are discarded. The finite ground set makes the remaining set of errors finite. The proposed constant is equivalently the limit of $-N^{-1}\log\delta(N)$. [S1] [S2]

Statement recorded in the supplied source.

Let $\delta(N)$ be the minimal non-zero value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$. Is it true that $ \delta(N)=e^{-(c+o(1))N} $ for some constant $c\in (0,1)$? [S1]

Short English statement

How close, without being equal, can a reciprocal subset sum from a finite interval come to one, and is the best error governed by one exponential rate?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2040 (EP-311), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 311, https://www.erdosproblems.com/311. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). Bibliographic information from the supplied record.