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2016 Equidistribution of values of linear forms on a cubic hypersurface
Sam Chow
Algebra Number Theory 10(2): 421-450 (2016). 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 $\alpha \in {ℝ}^{r}\setminus \left\{0\right\}$, 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 $h>16$.

## Citation

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Sam Chow. "Equidistribution of values of linear forms on a cubic hypersurface." Algebra Number Theory 10 (2) 421 - 450, 2016. https://doi.org/10.2140/ant.2016.10.421

## Information

Received: 29 April 2015; Revised: 9 November 2015; Accepted: 27 December 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06561469
MathSciNet: MR3477746
Digital Object Identifier: 10.2140/ant.2016.10.421

Subjects:
Primary: 11D25
Secondary: 11D75 , 11J13 , 11J71 , 11P55

Keywords: diophantine approximation , Diophantine equations , diophantine inequalities , equidistribution

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.10 • No. 2 • 2016