Erdős Problem #336

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.

The ordinary order is the least $r$ sufficient when up to $r$ summands are allowed; the exact order is the least $k$ sufficient with exactly $k$ summands. Repetitions of basis elements are allowed. Exact order $k$" must mean least such order, not merely an upper bound. [S1] [S2]

Statement recorded in the supplied source.

For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$). Find the value of $ \lim_r \frac{h(r)}{r^2}. $ [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

What is the asymptotic quadratic ratio between the largest finite exact order of a basis and its order when fewer summands are permitted?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2059 (EP-336), 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 336, https://www.erdosproblems.com/336. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207. Bibliographic information from the supplied record.

• S4 Grekos, Georges, Sur l'ordre d'une base additive. ([1988?]), Exp. No. 31, 13. Bibliographic information from the supplied record.

• S5 Nash, John C. M., Some applications of a theorem of {M}. {K}neser. J. Number Theory (1993), 1--8. Bibliographic information from the supplied record.

• S6 Plagne, Alain, À{} propos de la fonction {$X$} d'{E}rdős et {G}raham. Ann. Inst. Fourier (Grenoble) (2004), 1717--1767. Bibliographic information from the supplied record.