On the representation of quadratic forms by quadratic forms

TL;DR

This paper uses the circle method to establish when a fixed quadratic form can asymptotically represent large quadratic forms, depending on their dimensions and minima ratios.

Contribution

It provides new conditions under which the asymptotic formula for representing quadratic forms holds, extending previous results with precise dimension and size requirements.

Findings

Asymptotic formula holds for large enough forms

Conditions depend on dimensions and minima ratios

Results apply to positive definite integral quadratic forms

Abstract

Using the circle method, we show that for a fixed positive definite integral quadratic form $A$, the expected asymptotic formula for the number of representations of a positive definite integral quadratic form $B$ by $A$ holds true, providing that the dimension of $A$ is large enough in terms of the dimension of $B$ and the maximum ratio of the successive minima of $B$, and providing that $B$ is sufficiently large in terms of $A$.