Carbery's rectangle problem

Show that an open subset of the unit square with area α contains the corners of an axis-parallel rectangle of area at least cα².

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State the problem, from Ben Green's 100 open problems

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	1	+# Carbery's rectangle problem
	2	+
	3	+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.
	4	+
	5	+## Problem
	6	+
	7	+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
	8	+
	9	+$$(x_2 - x_1)(y_2 - y_1) \ge c\,\alpha^2.$$
	10	+
	11	+Equivalently: if a set contains the corners of no axis-parallel rectangle of area greater than $\lambda$, is its measure $O(\sqrt{\lambda})$?
	12	+
	13	+## Background
	14	+
	15	+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 $\|h_1 h_2\| \le N$?
	16	+
	17	+## What is known
	18	+
	19	+- 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 $\gg \alpha^2/\log(1/\alpha)$.
	20	+- A diagonal strip of width about $\alpha$ shows that $\alpha^2$ cannot be improved.
	21	+- 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 $\lambda$.
	22	+- 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.
	23	+- In three and more dimensions the corresponding box questions are wide open, even without any condition on volume.
	24	+
	25	+## What would count as progress
	26	+
	27	+- A proof of the $c\,\alpha^2$ bound, or any improvement of the logarithmic loss, such as $\alpha^2/(\log(1/\alpha))^{1-\varepsilon}$.
	28	+- 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.
	29	+- An answer to the discrete analogue above, for instance explicit sets for given $N$ whose additive quadruples can be checked exhaustively.
	30	+- A formal proof of the Lean statement `green_85` in [GreensOpenProblems/85.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/85.lean) (google-deepmind/formal-conjectures).
	31	+
	32	+## References
	33	+
	34	+1. A. Carbery, M. Christ and J. Wright, [Multidimensional van der Corput and sublevel set estimates](https://doi.org/10.1090/S0894-0347-99-00309-4), J. Amer. Math. Soc. 12 (1999), 981–1015.
	35	+2. N. H. Katz, [On the self crossing six sided figure problem](https://nyjm.albany.edu/j/1999/5-10.html), New York J. Math. 5 (1999), 121–130.
	36	+3. T. Keleti, [Density and covering properties of intervals of R^n](https://doi.org/10.1112/S0025579300015849), Mathematika 47 (2000), 229–242.
	37	+
	38	+## Source
	39	+
	40	+Imported from [Ben Green's 100 open problems 85](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.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.