Patterns without a popular difference
Name
2004.07722v2.pdf
Description
Published version
Size
433.87 KB
Format
Adobe PDF
Checksum (MD5)
d9855c347bb9991d338883490ea4b684
Author(s)
Sah, Ashwin
Sawhney, Mehtaab
Zhao, Yufei
Journal
Discrete Analysis, 2021:8, 30 pp
Publisher
Alliance of Diamond Open Access Journals
Citation
Sah, Ashwin, Sawhney, Mehtaab and Zhao, Yufei. "Patterns without a popular difference." Discrete Analysis, 2021:8, 30 pp, 2021.
Version
Final published version
Abstract
Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the
following property: for every $A \subseteq [N]^r$, there is some nonzero
integer $d$ such that $A$ contains $(\alpha^{|P|} - o(1))N^r$ translates of $d
\cdot P = \{d p : p \in P\}$, where $\alpha = |A|/N^r$?
Green showed that all 3-point $P \subseteq \mathbb{Z}$ have the above
property. Green and Tao showed that 4-point sets of the form $P = \{a, a+b,
a+c, a+b+c\} \subseteq \mathbb{Z}$ also have the property. We show that no
other sets have the above property. Furthermore, for various $P$, we provide
new upper bounds on the number of translates of $d \cdot P$ that one can
guarantee to find.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.19086/da.25317
