A truncated inner product formula in the geometry of numbers

Abstract

We study the statistical distribution of primitive sublattices in the space of lattices $\mathrm{SL}(n,\mathbb Z)\backslash\mathrm{SL}(n,\mathbb R)$. A central difficulty in this area is that the second moment of the counting function for rank $k$ sublattices, where $2 \le k \le n-2$, diverges. To overcome this, we analyze the inner product of truncated pseudo-Eisenstein series of the form $E_{f}(g) = \sum_{L} f(\det Lg)$, where the sum is over primitive rank $k$ sublattices of $\mathbb Z^n$. We establish an asymptotic formula for this inner product for both the standard Arthur truncation and a "harsh" truncation that vanishes outside a compact set. Our analysis relies on several technical results of independent interest, including a proof of the uniform moderate growth (UMG) property for these pseudo-Eisenstein series and a new method for resolving singularities in the Maass-Selberg relations. As a primary application, we obtain a significant improvement on the discrepancy bound for the number of primitive sublattices. For almost every lattice, we improve the error term in counting rank $k$ sublattices with determinant up to $p$ to $O(p^{n-1/7+ε})$, surpassing classical bounds for $\min(k, n-k) \ge 8$.