Erdős Problem #33

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.

Put $Q=\{m^2:m\in{\mathbb{Z}}_{\geq0}\}$ and $A(X)=|A\cap[1,X]|$. The admissible infinite sets $A$ satisfy $[N_0,\infty)\cap{\mathbb{N}}\subseteq Q+A$ for some $N_0$. The first question takes the infimum, over these fixed complements, of $\limsup_{X\to\infty}A(X)/\sqrt X$. [S1] [S2]

Statement recorded in the supplied source.

Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible value of $ \limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}? $ Is $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1? $ [S1]

Scope and interpretation.

The archive records a proved lower-limit bound. The optimization of the upper limiting square-root density is the remaining target. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine the smallest possible limsup A(x)/sqrt(x) for an additive complement to the squares. [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 smallest possible upper square-root density of a set whose translates by squares cover all sufficiently large integers?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1893 (EP-33), 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 33, https://www.erdosproblems.com/33. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Balasubramanian, R. and Ramana, D. S., Additive complements of the squares. C. R. Math. Acad. Sci. Soc. R. Can. (2001), 6--11. Bibliographic information from the supplied record.

• S4 Cilleruelo, Javier, The additive completion of {$k$}th-powers. J. Number Theory (1993), 237--243. Bibliographic information from the supplied record.

• S5 Habsieger, Laurent, On the additive completion of polynomial sets. J. Number Theory (1995), 130--135. Bibliographic information from the supplied record.

• S6 Moser, Leo, On the additive completion of sets of integers. (1965), 175--180. Bibliographic information from the supplied record.