#### 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
##### 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 $h>16$.

##### Keywords
diophantine equations, diophantine inequalities, diophantine approximation, equidistribution
##### Mathematical Subject Classification 2010
Primary: 11D25
Secondary: 11D75, 11J13, 11J71, 11P55
##### Milestones 