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On the polynomial Szemerédi theorem in finite fields. (English) Zbl 1473.11032

Summary: Let \(P_1,\ldots,P_m\in\mathbb{Z}[y]\) be any linearly independent polynomials with zero constant term. We show that there exists \(\gamma> 0\) such that any subset of \(\mathbb{F}_q\) of size at least \(q^{1-\gamma}\) contains a nontrivial polynomial progression \(x,x+P_{1}(y),\ldots,x+P_{m}(y)\), provided that the characteristic of \(\mathbb{F}_q\) is large enough.

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
11B25 Arithmetic progressions
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