<p>A skew corner is a triple of points in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z} \times \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo>×</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(x,y), (x, y + a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(x + a, y')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Pratt posed the following question: how large can a set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A \subseteq [n] \times [n]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> be, provided it contains no non-trivial skew corner (i.e. one for which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a\not =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>)? We prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$|A| \le \exp (- c\log ^c n) n^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mo>exp</mo> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>c</mi> <msup> <mo>log</mo> <mi>c</mi> </msup> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, for an absolute constant <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$c > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, which, along with a construction of Beker, essentially resolves Pratt’s question. Our argument represents a two-dimensional variant of the method of Kelley and Meka, which they used to prove Behrend-type bounds in Roth’s theorem. A very similar result was obtained independently and simultaneously by Jaber, Lovett and Ostuni.</p>

Good Bounds for Sets Lacking Skew Corners

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发表日期：

2026年1月7日

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Luka Milićević

摘要

A skew corner is a triple of points in $\mathbb {Z} \times \mathbb {Z}$ $Z \times Z$ of the form $(x,y), (x, y + a)$ $(x, y), (x, y + a)$ and $(x + a, y')$ $(x + a, y^{'})$ . Pratt posed the following question: how large can a set $A \subseteq [n] \times [n]$ $A \subseteq [n] \times [n]$ be, provided it contains no non-trivial skew corner (i.e. one for which $a\not =0$ $a \neq 0$ )? We prove that $|A| \le \exp (- c\log ^c n) n^2$ $| A | \leq exp (- c {log}^{c} n) n^{2}$ , for an absolute constant $c > 0$ $c > 0$ , which, along with a construction of Beker, essentially resolves Pratt’s question. Our argument represents a two-dimensional variant of the method of Kelley and Meka, which they used to prove Behrend-type bounds in Roth’s theorem. A very similar result was obtained independently and simultaneously by Jaber, Lovett and Ostuni.