Erdős Problem #870

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.

A minimal basis of order $k$ is a basis contained in $A$ that loses the at-most-$k$ basis property on removing any one of its elements. Representation counts must be taken with one consistent ordered or unordered convention; changing it only changes the allowed constant depending on fixed $k$. [S1] [S2]

Statement recorded in the supplied source.

Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.) [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Verify or refute the claimed construction; settle the exact-k formulation for k>=3. [S1]

Short English statement

Does sufficiently large logarithmic representation multiplicity ensure that an additive basis contains a subbasis from which no element can be removed?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2354 (EP-870), 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 870, https://www.erdosproblems.com/870. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, Paul and Nathanson, Melvyn B., Systems of distinct representatives and minimal bases in additive number theory. (1979), 89--107. Bibliographic information from the supplied record.

• S4 Härtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204. Bibliographic information from the supplied record.

• S5 Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333. Bibliographic information from the supplied record.