Abstract
Let
for any
Funding statement: The research is funded by the National Foundation for Science and Technology Development Project 101.99-2019.318.
References
[1] E. Aksoy Yazici, Sum-product type estimates for subsets of finite valuation rings, Acta Arith. 185 (2018), no. 1, 9–18. 10.4064/aa170418-12-12Search in Google Scholar
[2] D. N. V. Anh, L. Q. Ham, D. Koh, M. Mirzaei, H. Mojarrad and T. Pham, Moderate expanders over rings, J. Number Theory (2020), 10.1016/j.jnt.2020.07.009. 10.1016/j.jnt.2020.07.009Search in Google Scholar
[3] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra. Vol. 2, Addison-Wesley, Reading, 1969. Search in Google Scholar
[4]
M. Bennett, D. Hart, A. Iosevich, J. Pakianathan and M. Rudnev,
Group actions and geometric combinatorics in
[5] G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, Kluwer Int. Ser. Eng. Comput. Sci. 680, Kluwer Academic, Boston, 2002. 10.1007/978-1-4615-0957-8Search in Google Scholar
[6]
J. Bourgain,
The sum-product theorem in
[7] P. Erdős and E. Szemerédi, On sums and products of integers, Studies in Pure Mathematics, Birkhäuser, Basel (1983), 213–218. 10.1007/978-3-0348-5438-2_19Search in Google Scholar
[8] W. Fulton, Algebraic Curves, Adv. Book Class., Addison-Wesley, Redwood City, 1989. Search in Google Scholar
[9] L. Q. Ham, P. V. Thang and L. A. Vinh, Conditional expanding bounds for two-variable functions over finite valuation rings, European J. Combin. 60 (2017), 114–123. 10.1016/j.ejc.2016.09.009Search in Google Scholar
[10] D. Hart and A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464, American Mathematical Society, Providence (2008), 129–135. 10.1090/conm/464/09080Search in Google Scholar
[11] D. Hart, L. Li and C.-Y. Shen, Fourier analysis and expanding phenomena in finite fields, Proc. Amer. Math. Soc. 141 (2013), no. 2, 461–473. 10.1090/S0002-9939-2012-11338-3Search in Google Scholar
[12] P. D. Hiep, A note on sum-product estimates over finite valuation rings, preprint (2020), https://arxiv.org/abs/2005.05564. Search in Google Scholar
[13] B. Murphy, O. Roche-Newton and I. Shkredov, Variations on the sum-product problem, SIAM J. Discrete Math. 29 (2015), no. 1, 514–540. 10.1137/140952004Search in Google Scholar
[14] B. Nica, Unimodular graphs and Eisenstein sums, J. Algebraic Combin. 45 (2017), no. 2, 423–454. 10.1007/s10801-016-0712-8Search in Google Scholar
[15] T. Pham and L. A. Vinh, Some combinatorial number theory problems over finite valuation rings, Illinois J. Math. 61 (2017), no. 1–2, 243–257. 10.1215/ijm/1520046218Search in Google Scholar
[16] T. Pham, L. A. Vinh and F. de Zeeuw, Three-variable expanding polynomials and higher-dimensional distinct distances, Combinatorica 39 (2019), no. 2, 411–426. 10.1007/s00493-017-3773-ySearch in Google Scholar
[17] M. Rudnev, I. Shkredov and S. Stevens, On the energy variant of the sum-product conjecture, Rev. Mat. Iberoam. 36 (2020), no. 1, 207–232. 10.4171/rmi/1126Search in Google Scholar
[18] I. E. Shparlinski, On the solvability of bilinear equations in finite fields, Glasg. Math. J. 50 (2008), no. 3, 523–529. 10.1017/S0017089508004382Search in Google Scholar
[19] T. Tao, The sum-product phenomenon in arbitrary rings, Contrib. Discrete Math. 4 (2009), no. 2, 59–82. Search in Google Scholar
[20] L. A. Vinh, On four-variable expanders in finite fields, SIAM J. Discrete Math. 27 (2013), no. 4, 2038–2048. 10.1137/120892015Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
