Abstract
The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the Hardy– Littlewood circle method. There are a number of good texts available on the circle method, but the reader may particularly wish to study the books (Davenport, 2005) and (Vaughan, 1997).
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References
Birch, B. J. (1961/1962) Forms in many variables, Proc. Roy. Soc. Ser. A 265, 245–263.
Cook, R. J. (1971) Simultaneous quadratic equations, J. London Math. Soc. (2) 4, 319–326.
Davenport, H. (2005) Analytic methods for Diophantine equations and Diophantine inequalities, In Cambridge Mathematical Library, Cambridge, Cambridge University Press.
Davenport, H. and Lewis, D. J. (1969) Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A 264, 557–595.
Deligne, P. (1980) La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 52, 137–252.
Ford, K. (1995) New estimates for mean values of Weyl sums, Internat. Math. Res. Notices pp. 155–171.
Heath-Brown, D. R. (1996) A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481, 149–206.
Heath-Brown, D. R. (2003) Linear relations amongst sums of two squares, In London Math. Soc. Lecture Note Ser., Vol. 303 of London Math. Soc. Lecture Note Ser., Cambridge, pp. 133–176, Cambridge Univ. Press.
Heath-Brown, D. R. and Skorobogatov, A. N. (2002) Rational solutions of certain equations involving norms, Acta Math. 189, 161–177.
Hooley, C. (1994) On nonary cubic forms. III, J. Reine Angew. Math. 456, 53–63.
Vaughan, R. C. (1986) On Waring’s problem for cubes, J. Reine Angew. Math 365, 122–170.
Vaughan, R. C. (1997) The Hardy—Littlewood method, In Cambridge Tracts in Mathematics, Vol. 125, Cambridge-New York, Cambridge University Press.
Wang, Y. (1991) Diophantine equations and inequalities in algebraic number fields, Berlin, Springer-Verlag.
Weyl, H. (1916) Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77, 313–352.
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Heath-Brown, D. (2007). ANALYTIC METHODS FOR THE DISTRIBUTION OF RATIONAL POINTS ON ALGEBRAIC VARIETIES. In: Granville, A., Rudnick, Z. (eds) Equidistribution in Number Theory, An Introduction. NATO Science Series, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5404-4_8
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