Erdős Problem #875

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.

For each $r$, $S_r$ uses arbitrary $r$-element subsets of $A$, not just its first $r$ elements. Different lengths have disjoint sets of sums. This distinct-summand convention is essential: repetitions would create immediate unequal-length identities in positive integers. [S1] [S2]

Statement recorded in the supplied source.

Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite set such that the sets $ S_r = \{ a_1+\cdots +a_r : a_1<\cdots<a_r\in A\} $ are disjoint for distinct $r\geq 1$. How fast can such a sequence grow? How small can $a_{n+1}-a_n$ be? In particular, for which $c$ is it possible that $a_{n+1}-a_n\leq n^{c}$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine the optimal gap exponent and whether gaps below n are possible. [S1]

Short English statement

How small can successive gaps be in an infinite integer set whose distinct-element sums always determine the number of summands uniquely?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2357 (EP-875), 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 875, https://www.erdosproblems.com/875. Reference from the supplied record; the live problem page was not independently checked for this compilation.