Popular differences for right isosceles triangles
The electronic journal of combinatorics, Tome 28 (2021) no. 4
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.
@article{10_37236_10218,
author = {Vjekoslav Kova\v{c}},
title = {Popular differences for right isosceles triangles},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10218},
zbl = {1498.11041},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10218/}
}
Vjekoslav Kovač. Popular differences for right isosceles triangles. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10218
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