The positive-density refinement of the Fourier-coefficient cuspidality conjecture

Let $\mathscr U(\Gamma)\backslash\Lambda_n^+$ be the indexing set of positive-definite Fourier coefficients modulo the relevant unipotent subgroup, and let $S\subset\mathscr U(\Gamma)\backslash\Lambda_n^+$. Define the upper density

$$\delta_+(S):=\limsup_{X\to\infty}\frac{\#\{T\in S\mid\det(T)\leq X\}}{\#\{T\in\mathscr U(\Gamma)\backslash\Lambda_n^+\mid\det(T)\leq X\}}.$$

Positive-density refinement. A positive answer to the Fourier-coefficient growth conjecture holds when the growth condition is imposed on a set $S$ with $\delta_+(S)>0$. The paper contrasts this positive-density result with counterexamples for merely infinite thin sets; the general thin-set problem remains open.