Equidistribution of integers represented by standard quadratic form under arithmetic constraints
Yefei Ma
TL;DR
The paper addresses the local equidistribution of integers represented by the standard quadratic form $Q(x)=x_1^2+\cdots+x_d^2$ under arithmetic constraints modulo an odd prime $p$. It constructs weighted theta functions $\theta_f$ and shows they are modular (weight $d/2$) for appropriate congruence subgroups, with cusp-form criteria characterized by vanishing sums over $X_{p,d}(a)$ and $f(0)=0$. By bounding Fourier coefficients of cusp forms and establishing a precise singular-series framework for $r_d(n)$, the author proves that for $d\ge 4$, the representation set $X_d(n)$ becomes locally equidistributed on $p+1$ orbits as $n\to\infty$ (with explicit handling of $a\neq 0$ and $a=0$). This approach adapts Duke’s method for equidistribution on $S^2$ to higher dimensions via the Poisson-sum/cocycle formalism and the metaplectic framework, yielding quantitative, orbit-wise equidistribution results with implications for the distribution of lattice representations under arithmetic constraints.
Abstract
We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier expansion coefficients correspond to the counting problem over the quadric in $\mathbb{Z}^d$ induced by the standard quadratic form, subject to the aforementioned arithmetic constraints. The weak modular property of these modular forms allows us to use representation theory to identify the congruence subgroup to which our modular forms correspond. We then establish a necessary and sufficient condition for functions on $(\mathbb{Z}/p\mathbb{Z})^d$ that defines a cusp form. Finally, we conclude that the equidistribution phenomenon occurs locally on $p+1$ orbits for $d \geq 4$.