Equidistribution of values of linear forms on a cubic hypersurface

TL;DR

This paper establishes asymptotic formulas and equidistribution results for solutions of cubic hypersurfaces with linear forms, under certain algebraic independence and invariant conditions, advancing understanding of the distribution of solutions and their linear forms.

Contribution

The paper provides new asymptotic formulas for solutions to cubic hypersurfaces with linear forms and proves equidistribution of their linear form values under specific invariance conditions.

Findings

Asymptotic count of solutions with bounded variables

Equidistribution of linear form values at solutions

Relaxed conditions for existence of solutions

Abstract

Let $C$ be a cubic form with rational coefficients in $n$ variables, and let $h$ be the $h$-invariant of $C$. Let $L_1, \ldots, L_r$ be linear forms with real coefficients such that if $\boldsymbol{\alpha} \in \mathbb{R}^r \setminus \{ \boldsymbol{0} \}$ then $\boldsymbol{\alpha} \cdot \mathbf{L}$ is not a rational form. Assume that $h > 16 + 8 r$. Let $\boldsymbol{\tau} \in \mathbb{R}^r$, and let $\eta$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $\mathbf{x} \in [-P,P]^n$ to the system $C(\mathbf{x}) = 0, \: |\mathbf{L}(\mathbf{x}) - \boldsymbol{\tau}| < \eta$. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the $h$-invariant condition with the hypothesis $n > 16 + 9 r$, and show that the system has an integer solution. Finally, we show that the values of $\mathbf{L}$ at integer zeros of $C$ are equidistributed modulo one in $\mathbb{R}^r$, requiring only that $h > 16$.