Algebra & Number Theory

Equidistribution of values of linear forms on a cubic hypersurface

Sam Chow

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Abstract

Let C be a cubic form with integer coefficients in n variables, and let h be the h-invariant of C. Let L1,,Lr be linear forms with real coefficients such that, if α r {0}, then αL is not a rational form. Assume that h > 16 + 8r. Let τ r, and let η be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions x [P,P]n to the system C(x) = 0, |L(x) τ| < η. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the h-invariant condition with the hypothesis n > 16 + 9r and show that the system has an integer solution. Finally, we show that the values of L at integer zeros of C are equidistributed modulo 1 in r, requiring only that h > 16.

Article information

Source
Algebra Number Theory, Volume 10, Number 2 (2016), 421-450.

Dates
Received: 29 April 2015
Revised: 9 November 2015
Accepted: 27 December 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842483

Digital Object Identifier
doi:10.2140/ant.2016.10.421

Mathematical Reviews number (MathSciNet)
MR3477746

Zentralblatt MATH identifier
06561469

Subjects
Primary: 11D25: Cubic and quartic equations
Secondary: 11D75: Diophantine inequalities [See also 11J25] 11J13: Simultaneous homogeneous approximation, linear forms 11J71: Distribution modulo one [See also 11K06] 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

Keywords
diophantine equations diophantine inequalities diophantine approximation equidistribution

Citation

Chow, Sam. Equidistribution of values of linear forms on a cubic hypersurface. Algebra Number Theory 10 (2016), no. 2, 421--450. doi:10.2140/ant.2016.10.421. https://projecteuclid.org/euclid.ant/1510842483


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