Green’s Problem 24 on affine copies of {0,1,3}

For a finite set $A\subset\mathbb Z$ , let $M_{\{0,1,3\}}(A)$ be the number of pairs $(x,d)\in\mathbb Z\times(\mathbb Z\setminus\{0\})$ such that $\{x,x+d,x+3d\}\subseteq A$ . Determine the asymptotic extremal number of affine copies of $\{0,1,3\}$ in finite sets of integers; equivalently, determine the sharp constant governing $\max_{A\subset\mathbb Z,\,|A|=n}M_{\{0,1,3\}}(A)$ as $n\to\infty$ , or determine the corresponding limit superior $\limsup_{n\to\infty}\frac{1}{n^2}\max_{A\subset\mathbb Z,\,|A|=n}M_{\{0,1,3\}}(A)$ .

References

Primary source

arXiv

Additional references

Affine Copies of Three-Point Patterns in Sets of Integers — arXiv

additive-combinatorics extremal-combinatorics

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A September 2026 preprint improves the best known upper bounds, but does not determine the exact maximum, so the problem remains open.

The problem asks for the extremal number of affine copies of $\{0,1,3\}$ in a finite set of integers. The latest result treats this principal pattern within a broader family of three-point patterns.

September 2026 upper bounds

The preprint reports that, for coprime $\{0,a,b\}\ne\{0,1,2\}$ , the positive-dilation count is at most $\frac{99}{400}|A|^2+O(|A|)$ and the two-sided count at most $\frac{13}{28}|A|^2+O_P(|A|)$ . For $\{0,1,3\}$ specifically, it improves the bound to $\frac{47}{122}|A|^2+O(|A|)$ . These are upper bounds, not an exact extremal determination.

Current status (as of September 2026): The problem has a claimed new upper-bound improvement, but the exact extremal constant remains open.

Sources

No solutions have been posted yet.