Popular Differences for Right Isosceles Triangles
Keyword(s):
For a subset $A$ of $\{1,2,\ldots,N\}^2$ of size $\alpha N^2$ we show existence of $(m,n)\neq(0,0)$ such that the set $A$ contains at least $(\alpha^3 - o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers a question by Ackelsberg, Bergelson, and Best. The same approach also establishes the corresponding result for compact abelian groups. Furthermore, for a finite field $\mathbb{F}_q$ we comment on exponential smallness of subsets of $(\mathbb{F}_q^n)^2$ that avoid the aforementioned configuration. The proofs are minor modifications of the existing proofs regarding three-term arithmetic progressions.
2020 ◽
Vol 18
(04)
◽
pp. 2050019
1994 ◽
Vol 126
(1)
◽
pp. 1-6
◽
1966 ◽
Vol 62
(3)
◽
pp. 399-420
◽
Keyword(s):
2021 ◽
Vol 10
(4)
◽
pp. 2033-2044
1946 ◽
Vol 52
(10)
◽
pp. 940-945
◽
Keyword(s):
Keyword(s):