Geodesic
On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$
Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 625-634.

Voir la notice de l'article provenant de la source Math-Net.Ru

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, trigonometric sum, Cauchy–Bunyakovskii inequality.
Mots-clés : Fourier transform 
@article{MZM_2010_88_4_a13,
     author = {I. D. Shkredov},
     title = {On {Monochromatic} {Solutions} of {Some} {Nonlinear} {Equations} in~$\mathbb Z/p\mathbb Z$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {625--634},
     publisher = {mathdoc},
     volume = {88},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a13/}
}
TY  - JOUR
AU  - I. D. Shkredov
TI  - On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$
JO  - Matematičeskie zametki
PY  - 2010
SP  - 625
EP  - 634
VL  - 88
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a13/
LA  - ru
ID  - MZM_2010_88_4_a13
ER  - 
%0 Journal Article
%A I. D. Shkredov
%T On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$
%J Matematičeskie zametki
%D 2010
%P 625-634
%V 88
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a13/
%G ru
%F MZM_2010_88_4_a13
I. D. Shkredov. On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$. Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 625-634. http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a13/

[1] R. L. Graham, B. L. Rothschild, J. H. Spencer, Ramsey Theory, Wiley-Intersci. Ser. Discrete Math. Optim., John Wiley Sons, New York, 1990 | MR | Zbl

[2] R. Rado, “Verallgemeinerung eines Satzes von van der Waerden mit Anwendungen auf ein Problem der Zahlentheorie”, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., 17 (1933), 589–596 | Zbl

[3] R. Rado, “Studien zur Kombinatorik”, Math. Z., 36:1 (1933), 424–470 | DOI | MR | Zbl

[4] R. Rado, “Some recent results in combinatorial analysis”, Comptes Rendus du Congres International des Mathematiciens, v. 2, Oslo, 1937, 20–21 | Zbl

[5] W. Deuber, “Partitionen und lineare Gleichungssysteme”, Math. Z., 133:2 (1973), 109–123 | DOI | MR | Zbl

[6] W. Deuber, “Partitions theorems for Abelian groups”, J. Combin. Theory Ser. A, 19:1 (1975), 95–108 | DOI | MR | Zbl

[7] I. Schur, “Über die Kongruenz $x^m+y^m\equiv z^m\pmod p$”, Deutsche Math. Ver., 25 (1916), 114–117 | Zbl

[8] N. Hindman, I. Leader, D. Strauss, “Open problems in partition regularity”, Combin. Probab. Comput., 12 (2003), 571–583 | DOI | MR | Zbl

[9] M. Beiglböck, V. Bergelson, N. Hindman, D. Strauss, “Multiplicative structures in additively large sets”, J. Combin. Theory Ser. A, 113:7 (2006), 1219–1242 | DOI | MR | Zbl

[10] M. Beiglböck, V. Bergelson, N. Hindman, D. Strauss, “Some new results in multiplicative and additive Ramsey theory”, Trans. Amer. Math. Soc., 360:2 (2008), 819–847 | MR | Zbl

[11] V. Bergelson, N. Hindman, I. Leader, “Additive and multiplicative Ramsey theory in the reals and the rationals”, J. Combin. Theory Ser. A, 85:1 (1999), 41–68 | DOI | MR | Zbl

[12] S. D. Adhikari, “A note on a question of Erdős”, Exposition. Math., 15:4 (1997), 367–371 | MR | Zbl

[13] T. C. Brown, V. Rödl, “Monochromatic solutions to equations with unit fractions”, Bull. Austral. Math. Soc., 43:3 (1991), 387–392 | DOI | MR | Zbl

[14] H. Lefmann, “On partition regular systems of equations”, J. Combin. Theory Ser. A, 58:1 (1991), 35–53 | DOI | MR | Zbl

[15] I. E. Shparlinski, On The Solvability of Bilinear Equations in Finite Fields, arXiv: math.NT/0708.2130

[16] A. Sárközy, “On sums and products of residues modulo $p$”, Acta Arith., 118:4 (2005), 403–409 | DOI | MR | Zbl

[17] M. Z. Garaev, V. Garcia, The Equation $x_1x_2=x_3x_4+\lambda$ in Fields of Prime Order and Applications, Preprint, 2007

[18] J. Bourgain, More on the Sum-Product Phenomenon in Prime Fields and its Applications, Preprint

[19] N. Hegyváry, F. Hennecart, Explicit Constructions of Extractors and Expanders, Preprint

[20] M. Z. Garaev, C.-Y. Shen, On the Size of the Set $A(A+1)$, arXiv: math.NT/0811.4206

[21] S. V. Konyagin, I. E. Shparlinski, Character Sums with Exponential Functions and their Applications, Cambridge Tracts in Math., 136, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[22] J. Johnsen, “On the distibution of powers in finite fields”, J. Reine Angew. Math., 251 (1971), 10–19 | MR | Zbl

[23] I. M. Vinogradov, Osnovy teorii chisel, Lan, SPb., 2004 | MR | Zbl

[24] W. T. Gowers, “A new proof of Szemerédi's theorem for arithmetic progressions of length four”, Geom. Funct. Anal., 8:3 (1998), 529–551 | MR | Zbl

[25] W. T. Gowers, “A new proof of Szemerédi's theorem”, Geom. Funct. Anal., 11:3 (2001), 465–588 | MR | Zbl