Equidistribution of values of linear forms on a cubic hypersurface

Chow, Sam

doi:10.2140/ant.2016.10.421

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Equidistribution of values of linear forms on a cubic hypersurface

Sam Chow

Vol. 10 (2016), No. 2, 421–450

DOI: 10.2140/ant.2016.10.421

Abstract

Let $C$ be a cubic form with integer coefficients in $n$ variables, and let $h$ be the $h$ -invariant of $C$ . Let $L_{1}, \dots, L_{r}$ be linear forms with real coefficients such that, if $α \in ℝ^{r} ∖ {0}$ , then $α \cdot L$ is not a rational form. Assume that $h > 16 + 8 r$ . Let $τ \in ℝ^{r}$ , and let $η$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $x \in {[- P, P]}^{n}$ 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 $n > 16 + 9 r$ 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 $ℝ^{r}$ , requiring only that $h > 16$ .

Keywords

diophantine equations, diophantine inequalities, diophantine approximation, equidistribution

Mathematical Subject Classification 2010

Primary: 11D25

Secondary: 11D75, 11J13, 11J71, 11P55

Milestones

Received: 29 April 2015

Revised: 9 November 2015

Accepted: 27 December 2015

Published: 16 March 2016

Authors

Sam Chow
School of Mathematics University of Bristol University Walk Bristol BS8 1TW United Kingdom

Vol. 10, No. 2, 2016