Abstract
In earlier papers Sárközy studied the solvability of the equations
resp.
where \(\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}\) are “large” subsets of \(\mathbb{F}_p\). Later Gyarmati and Sárközy generalized and extended these problems by studying these equations and also other algebraic equations with restricted solution sets over finite fields. Here we will continue the work by studying further special equations over finite fields and also algebraic equations with restricted solution sets over the set of the integers, resp. rationals. We will focus on the most interesting cases of algebraic equations with 3, resp. 4 variables. In the cases when there are no “density results” of the above type, we will be also looking for Ramsey type results, i.e., for monochromatic solutions of the given equation. While in the earlier papers character sum estimates were used, now combinatorial tools dominate.
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References
F. Behrend: On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331–332.
J. Bourgain: On triples in arithmetic progressions, Geom. Funct. Anal. 9 (1999), 968–984.
P. Erdös and A. Sárközy: On a conjecture of Roth and some related problems, II, in: Number Theory, Proceedings of the First Conference of the Canadian Number Theory Association (held at Banff Center, Banff, Alberta, April 17–27, 1988), ed. R. A. Mollin, Walter de Gruyter, Berlin-New York, 1990; 125–138.
P. Erdös, A. Sárközy and V. T. Sós: On a conjecture of Roth and some related problems, I, in: Irregularities of Partitions, eds. G. Halász and V. T. Sós, Algorithms and Combinatorics 8, Springer-Verlag, Berlin-Heidelberg-New York, 1989; 47–59.
K. Gyarmati: On a problem of Diophantus, Acta Arith. 97 (2001), 53–65.
K. Gyarmati and A. Sárközy: Equations in finite fields with restricted solution sets, I (Character sums), Acta Math. Hungar. 118 (2008), 129–148.
K. Gyarmati and A. Sárközy: Equations in finite fields with restricted solution sets, II (Algebraic equations), Acta Math. Hungar. 119 (2008), 259–280.
N. Hindman: Monochromatic sums equal to products in N, Integers 11A (2011), Article 10, 10.
K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.
A. Sárközy: On sums and products of residues modulo p, Acta Arith. 118 (2005), 403–409.
A. Sárközy: On products and shifted products of residues modulo p, Integers 8 (2008), Article 9, 8.
J. Schur: Über die Kongruenz x m+y m≡z m (mod p), Jahresber. Deutschen Math. Verein. 25 (1916), 114–117.
B. L. van der Waerden: Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212–216.
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Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631, T 049693 and PD72264 and the János Bolyai Research Fellowship.
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Csikvári, P., Gyarmati, K. & Sárközy, A. Density and ramsey type results on algebraic equations with restricted solution sets. Combinatorica 32, 425–449 (2012). https://doi.org/10.1007/s00493-012-2697-9
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DOI: https://doi.org/10.1007/s00493-012-2697-9