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arXiv · 1811.09947

On arithmetic progressions in symmetric sets in finite field model

Jan Hązła
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Abstract

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 $S\subseteq\mathbb{F}^n_p$ 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.

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Jan Hązła. 2018-11-25. On arithmetic progressions in symmetric sets in finite field model. https://arxiv.org/abs/1811.09947

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