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Lower bounds for corner-free sets

Abstract:
A corner is a set of three points in $\mathbf{Z}^2$ of the form $(x, y), (x + d, y), (x, y + d)$ with $d \neq 0$. We show that for infinitely many $N$ there is a set $A \subset [N]^2$ of size $2^{-(c + o(1)) \sqrt{\log_2 N}} N^2$ not containing any corner, where $c = 2 \sqrt{2 \log_2 \frac{4}{3}} \approx 1.822\dots$.
Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.53733/86

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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Oxford college:
Magdalen College
Role:
Author
ORCID:
0000-0002-2224-1193
Publisher:
New Zealand Mathematical Society
Journal:
New Zealand Journal of Mathematics More from this journal
Volume:
51
Pages:
1–2
Publication date:
2021-07-29
Acceptance date:
2021-03-08
DOI:
EISSN:
1179-4984
Language:
English
Keywords:
Pubs id:
1164150
Local pid:
pubs:1164150
Deposit date:
2023-03-03

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