Carbery's rectangle problem

A set covering a fraction $\alpha$ of the unit square must contain the four corners of some axis-parallel rectangle, but how large must that rectangle be? A thin diagonal strip shows that one cannot hope for more than about $\alpha^2$, a short Cauchy–Schwarz argument guarantees $\alpha^2/\log(1/\alpha)$, and closing this logarithmic gap is open. The question comes from sublevel-set estimates in harmonic analysis.

Problem

Is there an absolute constant $c > 0$ with the following property? For every open set $A \subseteq [0, 1]^2$ of Lebesgue measure $\alpha > 0$ there are $x_1 < x_2$ and $y_1 < y_2$ such that all four points $(x_1, y_1), (x_1, y_2), (x_2, y_1), (x_2, y_2)$ lie in $A$ and

$$(x_2 - x_1)(y_2 - y_1) \ge c\,\alpha^2.$$

Equivalently: if a set contains the corners of no axis-parallel rectangle of area greater than $\lambda$, is its measure $O(\sqrt{\lambda})$?

Background

The problem appears in Section 6 of the paper of Carbery, Christ and Wright (1999) on multidimensional van der Corput lemmas and sublevel set estimates, where the bound with a logarithmic loss is proved; it is often called Carbery's rectangle problem. Sublevel set estimates bound the measure of the set where a function of several variables is small, and they underlie multidimensional versions of van der Corput's lemma for oscillatory integrals. Keleti (2000) studied related density and covering properties of intervals in $\mathbb{R}^n$. Green also poses a discrete analogue: is there $A \subseteq \{1, \dots, N\}$ with $|A| \ge 100\sqrt{N}$ such that whenever $x, x + h_1, x + h_2, x + h_1 + h_2$ all lie in $A$ one has $|h_1 h_2| \le N$?

What is known

• Carbery, Christ and Wright (1999): a measurable set in the unit square with no rectangle corners of area greater than $\lambda$ has measure $O(\sqrt{\lambda \log(1/\lambda)})$; equivalently a set of measure $\alpha$ contains corners of a rectangle of area $\gg \alpha^2/\log(1/\alpha)$.
• A diagonal strip of width about $\alpha$ shows that $\alpha^2$ cannot be improved.
• Katz (1999) removed the logarithm under the extra assumption that the set contains no corners of an axis-parallel, possibly self-crossing hexagon of area greater than $\lambda$.
• If one instead asks for a rectangle $R$ with all four corners in $A$ and $|A \cap R| \gg \alpha^C$, this is known with $C = 4$ but fails with $C = 3$, by an example Keleti attributes to Reiman.
• In three and more dimensions the corresponding box questions are wide open, even without any condition on volume.

What would count as progress

• A proof of the $c\,\alpha^2$ bound, or any improvement of the logarithmic loss, such as $\alpha^2/(\log(1/\alpha))^{1-\varepsilon}$.
• A counterexample family: open sets of measure $\alpha$ in which every corner rectangle has area $o(\alpha^2)$. For a finite union of grid boxes the largest corner rectangle can be computed exactly, so candidate constructions can be tested by computer.
• An answer to the discrete analogue above, for instance explicit sets for given $N$ whose additive quadruples can be checked exhaustively.
• A formal proof of the Lean statement green_85 in GreensOpenProblems/85.lean (google-deepmind/formal-conjectures).

References

1. A. Carbery, M. Christ and J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), 981–1015.
2. N. H. Katz, On the self crossing six sided figure problem, New York J. Math. 5 (1999), 121–130.
3. T. Keleti, Density and covering properties of intervals of R^n, Mathematika 47 (2000), 229–242.

Source

Imported from Ben Green's 100 open problems 85, curated by Ben Green. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.