A refinement of Heath-Browns theorem on quadratic forms

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Abstract

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be C∞0-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not to vanish at the singularity and have an explicit decay at infinity.
The paper uses only elementary number theory and is available to readers with no number-theoretic background.

About the authors

Sergei Georgievich Vlăduţ

Aix-Marseille Université; Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)

Email: vladut@iml.univ-mrs.fr
Candidate of physico-mathematical sciences

Andrey Victorovich Dymov

Steklov Mathematical Institute of Russian Academy of Sciences; HSE University; Skolkovo Institute of Science and Technology

Email: dymov@mi-ras.ru
PhD, no status

Sergei Borisovich Kuksin

Paris Sorbonne University; Peoples' Friendship University of Russia; Steklov Mathematical Institute of Russian Academy of Sciences

Author for correspondence.
Email: vladut@iml.univ-mrs.fr

Doctor of physico-mathematical sciences, Professor

Alberto Maiocchi

Università degli Studi di Milano-Bicocca

Email: alberto.maiocchi@unimib.it

References

  1. D. R. Heath-Brown, “A new form of the circle method, and its application to quadratic forms”, J. Reine Angew. Math., 1996:481 (1996), 149–206
  2. W. Duke, J. Friedlander, H. Iwaniec, “Bounds for automorphic $L$-function”, Invent. Math., 112:1 (1993), 1–8
  3. H. Iwaniec, “The circle method and the Fourier coefficients of modular forms”, Number theory and related topics (Bombay, 1988), Tata Inst. Fund. Res. Stud. Math., 12, Tata Inst. Fund. Res., Bombay, 1989, 47–55
  4. Ж.-П. Серр, Курс арифметики, Мир, М., 1972, 184 с.
  5. А. А. Карацуба, Основы аналитической теории чисел, 2-е изд., Наука, М., 1983, 240 с.
  6. A. Dymov, S. Kuksin, A. Maiocchi, S. Vlăduţ, The large-period limit for equations of discrete turbulence
  7. T. Buckmaster, P. Germain, Z. Hani, J. Shatah, “Effective dynamics of the nonlinear Schrödinger equation on large domains”, Comm. Pure Appl. Math., 71:7 (2018), 1407–1460
  8. H. L. Eliasson, B. Grebert, S. B. Kuksin, “KAM for the nonlinear beam equation”, Geom. Funct. Anal., 26:6 (2016), 1588–1715
  9. J. R. Getz, “Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields”, J. Lond. Math. Soc. (2), 98:2 (2018), 275–305
  10. T. H. Tran, Secondary terms in asymptotics for the number of zeros of quadratic forms
  11. A. Dymov, S. Kuksin, “Formal expansions in stochastic model for wave turbulence. 1: Kinetic limit”, Comm. Math. Phys., 382:2 (2021), 951–1014
  12. I. Chavel, Riemannian geometry. A modern introduction, Cambridge Stud. Adv. Math., 98, 2nd ed., Cambridge Univ. Press, Cambridge, 2006, xvi+471 pp.
  13. A. Dymov, S. Kuksin, A. Maiocchi, S. Vlăduţ, Some remarks on Heath-Brown's theorem on quadratic forms
  14. А. Я. Хинчин, Математические основания статистической механики, Гостехиздат, М.–Л., 1943, 128 с.
  15. B. J. Birch, “Forms in many variables”, Proc. Roy. Soc. London Ser. A, 265:1321 (1962), 245–263

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Copyright (c) 2023 Vlăduţ S.G., Dymov A.V., Kuksin S.B., Maiocchi A.

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