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ON THE SOLVABILITY OF SYSTEMS OF SUM–PRODUCT EQUATIONS IN FINITE FIELDS

Published online by Cambridge University Press:  01 August 2011

LE ANH VINH
LE ANH VINH*
Affiliation:
Mathematics Department, Harvard University, Cambridge, MA 02138, USA e-mail: [email protected]
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Abstract

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In an earlier paper, for ‘large’ (but otherwise unspecified) subsets , , , of q, Sárközy showed the solvability of the equations a + b = cd with a ∈ , b ∈ , c ∈ , d ∈ . This equation has been studied recently by many other authors. In this paper, we study the solvability of systems of equations of this type using additive character sums.

Keywords

05C3505C3805C55
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 427 - 435
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Chapman, J., Erdogan, M. B., Hart, D., Iosevich, A. and Koh, D., Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates, Math. Zeitschr. (to appear).Google Scholar
2.Covert, D., Hart, D., Iosevich, A. and Uriarte-Tuero, I., An analog of the Furstenberg-Katznelson–Weiss theorem on triangles in sets of positive density in finite field geometries, Discr. Math. (to appear).Google Scholar
3.Garaev, M. Z., The sum-product estimate for large subsets of prime fields, Proc. Amer. Math. Soc. 136 (2008), 2735–2739.CrossRefGoogle Scholar
4.Gyarmati, K. and Sárközy, A., Equations in finite fields with restricted solution sets, II (algebraic equations), Acta Math. Hungar. 119 (2008), 259–280.CrossRefGoogle Scholar
5.Hart, D. and Iosevich, A., Ubiquity of simplices in subsets of vector spaces over finite fields, Anal. Math. 34 (2007), 29–38.CrossRefGoogle Scholar
6.Hart, D. and Iosevich, A., Sums and products in finite fields: an integral geometric viewpoint, Contemp. Math. 464 (2008).CrossRefGoogle Scholar
7.Hart, D., Iosevich, A., Koh, D. and Rudnev, M., Averages over hyperplanes, sum–product theory in finite fields, and the Erdös-Falconer distance conjecture, Trans. AMS 363 (2011) 3255–3275.CrossRefGoogle Scholar
8.Sárközy, A., On products and shifted products of residues modulo p, Integers: Electron. J. Combin. Numb. Theory 8 (2) (2008), A9, 1–8.Google Scholar
9.Shparlinski, I. E., On the solvability of bilinear equations in finite fields, Glassgow Math. J. 50 (2008), 523–529.CrossRefGoogle Scholar
10.Vinh, L. A., On the solvability of bilinear equations in finite fields, Proc. Amer. Math. Soc. 137 (2009), 2889–2898.CrossRefGoogle Scholar
11.Vinh, L. A., Szemerédi–Trotter type theorem and sum–product estimate in finite fields, Eur. J. Combin., accepted.Google Scholar
12.Vinh, L. A., On a Furstenberg-Katznelson-Weiss type theorem over finite fields, Ann. Combin. (to appear).Google Scholar
13.Vinh, L. A., Triangles in vector spaces over finite fields, Online J. Anal. Combin. (to appear).Google Scholar
14.Vinh, L. A., On kaleidoscopic pseudorandomness of finite Euclidean graphs, Discuss. Math. Graph Theory (to appear).Google Scholar
15.Vinh, L. A., On some problems of Sárközy and Gyarmati, Integers: Electron. J. Combin. Numb. Theory (to appear).Google Scholar