Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-11T02:16:14.213Z Has data issue: false hasContentIssue false
  • English
  • Français

On multiplicative energy of subsets of varieties

Part of: Harmonic analysis in several variables Designs and configurations Sequences and sets Linear algebraic groups and related topics

Published online by Cambridge University Press:  13 January 2022

Ilya D. Shkredov
Ilya D. Shkredov*
Affiliation:
Department of Number Theory, Steklov Mathematical Institute, ul. Gubkina, 8, Moscow 119991, Russia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.

Keywords

Multiplicative energyvarietiesrestriction phenomenon

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11B75: Other combinatorial number theory 11B99: None of the above, but in this section 05B99: None of the above, but in this section 20G15: Linear algebraic groups over arbitrary fields 42B05: Fourier series and coefficients
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 322 - 340
Check for updates
Copyright
© Canadian Mathematical Society, 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work is supported by the Russian Science Foundation under grant 19-11-00001.

References

Bollobás, B. and Thomason, A., Projections of bodies and hereditary properties of hypergraphs. Bull. Lond. Math. Soc. 27(1995), 417–424.CrossRefGoogle Scholar
Bourgain, J., Some new estimates on oscillatory integrals. In: Fefferman, C., Fefferman, R., and Wainger, S. (eds.), Essays in Fourier analysis in honor of E. M. Stein, Princeton University Press, 1995, pp. 83–112.Google Scholar
Bourgain, J., Harmonic analysis and combinatorics: how much may they contribute to each other? In: Mathematics: Frontiers and perspectives, IMU/Amer. Math. Society, 2000, pp. 13–32.Google Scholar
Bourgain, J., Gamburd, A., and Sarnak, P., Affine linear sieve, expanders, and sum–product. Invent. Math. 179(2010), no. 3, 559–644.CrossRefGoogle Scholar
Breuillard, E., Green, B., and Tao, T., Approximate subgroups of linear groups. Geom. Funct. Anal. 21(2011), no. 4, 774–819.CrossRefGoogle Scholar
Gowers, W. T., A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11 (2001), 465–588.CrossRefGoogle Scholar
Gowers, W. T., Quasirandom groups. Probab. Comput. 17(2008), no. 3, 363–387.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, 52, Springer Science & Business Media, New York, 2013.Google Scholar
Heintz, J., Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci. 24(1983), 239–277.CrossRefGoogle Scholar
Helfgott, H., Growth and generation in ${SL}_2\left(\mathbb{Z}/p\mathbb{Z}\right)$ . Ann. of Math. (2) 167(2008), no. 2, 601–623.CrossRefGoogle Scholar
Helfgott, H., Growth in groups: ideas and perspectives. Bull. Amer. Math. Soc. (N.S.) 52(2015), no. 3, 357–413.CrossRefGoogle Scholar
Humphreys, J. E., Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43, American Mathematical Society, Providence, RI, 2011.CrossRefGoogle Scholar
Iosevich, A. and Koh, D., Extension theorems for spheres in the finite field setting. Forum Math. 22(2010), no.3, 457–483.CrossRefGoogle Scholar
Iosevich, A., Koh, D., and Lewko, M., Finite field restriction estimates for the paraboloid in high even dimensions. J. Funct. Anal. 278(2020), 108450.CrossRefGoogle Scholar
Landazuri, V. and Seitz, G. M., On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32(1974), 418–443.CrossRefGoogle Scholar
Lang, S. and Weil, A., Number of points of varieties in finite fields. Amer. J. Math. 76(1954), 819–827.CrossRefGoogle Scholar
Larsen, M. J. and Pink, R., Finite subgroups of algebraic groups. J. Amer. Math. Soc. 24(2011), no. 4, 1105–1158.CrossRefGoogle Scholar
Lewko, M., Finite field restriction estimates based on Kakeya maximal operator estimates. J. Eur. Math. Soc. (JEMS) 21(2019), no. 12, 3649–3707.CrossRefGoogle Scholar
Liebeck, M. W., Schul, G., and Shalev, A., Rapid growth in finite simple groups. Trans. Amer. Math. Soc. 369(2017), no. 12, 8765–8779.CrossRefGoogle Scholar
Liebeck, M. W. and Shalev, A., Diameters of finite simple groups: sharp bounds and applications. Ann. of Math. (2) 154(2001), 383–406.CrossRefGoogle Scholar
Mockenhaupt, G. and Tao, T., Restriction and Kakeya phenomena for finite fields. Duke Math. J. 121(2004), no. 1, 35–74.CrossRefGoogle Scholar
Murphy, B., Upper and lower bounds for rich lines in grids. Amer. J. Math. 143 (2021), no. 2, 577–611.Google Scholar
Naimark, M. A., Theory of group representations, Fizmatlit, Moscow, 2010.Google Scholar
Noether, E., Ein algebraisches Kriterium für absolute Irreduzibilität. Math. Ann. 85(1922), 26–40.CrossRefGoogle Scholar
Pyber, L. and Szabó, E., Growth in finite simple groups of Lie type. J. Amer. Math. Soc. 29(2016), no. 1, 95–146.CrossRefGoogle Scholar
Sanders, T., A quantitative version of the non-abelian idempotent theorem. Geom. Funct. Anal. 21(2011), no. 1, 141–221.CrossRefGoogle Scholar
Serre, J. P., Représentations linéaires des groupes finis, Collections Méthodes, Hermann, Paris, 1967.Google Scholar
Shkredov, I. D., Energies and structure of additive sets. Electron. J. Combin. 21(2014), no. 3, #P3.44, 1–53.Google Scholar
Shkredov, I. D., On asymptotic formulae in some sum–product questions. Trans. Moscow Math. Soc. 79(2018), 271–334; English transl. Trans. Moscow Math. Soc. (2018), 231–281.CrossRefGoogle Scholar
Shkredov, I. D., Growth in Chevalley groups relatively to parabolic subgroups and some applications. Preprint, 2020. arXiv:2003.12785 Google Scholar
Shkredov, I. D., Modular hyperbolas and bilinear forms of Kloosterman sums. J. Number Theory 220(2021), 182–211.CrossRefGoogle Scholar
Shkredov, I. D., On the spectral gap and the diameter of Cayley graphs. Proc. Steklov Inst. Math. 314(2021), 307–324.CrossRefGoogle Scholar
Stein, E. M., Some problems in harmonic analysis . In: Harmonic analysis in Euclidean spaces, Proceedings of Symposia in Pure Mathematics, XXXV Part I, Williams College, Williamstown, MA, 1978, pp. 3–20.Google Scholar
Stein, E. M., Harmonic analysis, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Tao, T. and Vu, V., Additive combinatorics, Cambridge University Press, Cambridge, UK, 2006.CrossRefGoogle Scholar
Volobuev, A., Preprint.Google Scholar