Equidistribution of values of linear forms on a cubic hypersurface

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 $\alpha \in {ℝ}^r\setminus \left\{0\right\}$, then $\alpha \cdot L$ is not a rational form. Assume that $h>16+8r$. Let $\tau \in {ℝ}^r$, and let $\eta$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $x\in {\left[-P,P\right]}^n$ to the system $C\left(x\right)=0$, $|L\left(x\right)-\tau |<\eta$. 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 ${ℝ}^r$, requiring only that $h>16$.