On arithmetic progressions in symmetric sets in finite field model
The electronic journal of combinatorics, Tome 27 (2020) no. 3
We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S \subseteq \mathbb{Z}_q^n$ containing $|S| = \mu \cdot q^n$ elements must contain at least $\delta(q, \mu) \cdot q^n \cdot 2^n$ arithmetic progressions $x, x+d, \ldots, x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}_p^n$ with $|S|=\mu\cdot p^n$ elements contains at least $\mu^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.
@article{10_37236_9242,
author = {Jan H\k{a}z{\l}a},
title = {On arithmetic progressions in symmetric sets in finite field model},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {3},
doi = {10.37236/9242},
zbl = {1468.11038},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9242/}
}
Jan Hązła. On arithmetic progressions in symmetric sets in finite field model. The electronic journal of combinatorics, Tome 27 (2020) no. 3. doi: 10.37236/9242
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