Covering Squares with Sumsets

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite set $A$ of real numbers, write $A+A=\{a+b:a,b\in A\}$; repeated summands are allowed. The first $n$ positive squares are $1^2,2^2,\ldots,n^2$. The asymptotic lower bound $n^{1-o(1)}$ means a lower bound $n^{1-\epsilon}$ for every fixed $\epsilon>0$, once $n$ is sufficiently large depending only on $\epsilon$.

The exponent question is equivalent if $A$ is required to consist of integers. Indeed, for arbitrary real $A$, the integer set $B=\{\lfloor a\rfloor,\lceil a\rceil:a\in A\}$ has $|B|\leq2|A|$, and every integer belonging to $A+A$ belongs to $B+B$. A constant factor of two does not affect the exponent $1-o(1)$.

Statement recorded in the supplied source.

For every $\epsilon>0$, does there exist $n_0(\epsilon)$ such that for all integers $n\geq n_0(\epsilon)$ and all finite $A\subset\mathbb R$, \[ \{1^2,2^2,\ldots,n^2\}\subset A+A \quad\Longrightarrow\quad |A|\geq n^{1-\epsilon}? \] [S2]

Short English statement

Must a set whose pairwise sums contain every one of the first n positive squares have cardinality nearly n on the exponent scale?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 122 (GREEN-034). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 61 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.