Vol. 10, No. 2, 2016

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Equidistribution of values of linear forms on a cubic hypersurface

Sam Chow

Vol. 10 (2016), No. 2, 421–450

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.

diophantine equations, diophantine inequalities, diophantine approximation, equidistribution
Mathematical Subject Classification 2010
Primary: 11D25
Secondary: 11D75, 11J13, 11J71, 11P55
Received: 29 April 2015
Revised: 9 November 2015
Accepted: 27 December 2015
Published: 16 March 2016
Sam Chow
School of Mathematics
University of Bristol
University Walk
United Kingdom