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Mat. Zametki, 2010, Volume 88, Issue 4, Pages 625–634 (Mi mz6581)  

This article is cited in 12 scientific papers (total in 12 papers)

On Monochromatic Solutions of Some Nonlinear Equations in $\mathbb Z/p\mathbb Z$

I. D. Shkredov

M. V. Lomonosov Moscow State University

Abstract: Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are $x,y\in\mathbb Z$ such that $x+y$, $xy$, and $x$ have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group $\mathbb Z/p\mathbb Z$, where $p$ is a prime, and obtain an even stronger density result.

Keywords: Ramsey theory, coloring, monochromatic solution, Dirichlet character, Fourier transform, trigonometric sum, Cauchy–Bunyakovskii inequality

DOI: https://doi.org/10.4213/mzm6581

Full text: PDF file (504 kB)
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English version:
Mathematical Notes, 2010, 88:4, 603–611

Bibliographic databases:

UDC: 514.7
Received: 22.12.2009

Citation: I. D. Shkredov, “On Monochromatic Solutions of Some Nonlinear Equations in $\mathbb Z/p\mathbb Z$”, Mat. Zametki, 88:4 (2010), 625–634; Math. Notes, 88:4 (2010), 603–611

Citation in format AMSBIB
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    3. Le Anh Vinh, “On Four-Variable Expanders in Finite Fields”, SIAM Discret. Math., 27:4 (2013), 2038–2048  crossref  mathscinet  zmath  isi  scopus
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    6. Petrov F., “Combinatorial Nullstellensatz Approach To Polynomial Expansion”, Acta Arith., 165:3 (2014), 279–282  crossref  mathscinet  zmath  isi  scopus
    7. Bergelson V., Moreira J., “Ergodic theorem involving additive and multiplicative groups of a field and patterns”, Ergod. Theory Dyn. Syst., 37:3 (2017), 673–692  crossref  mathscinet  zmath  isi  scopus
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