2.3 Fixed-\(n\) asymptotics
For every fixed integer \(n\ge 2\), \(N_{n,4t}=[1+o_{t\to \infty }(1)]\, A_{n,4t}\) as \(t\to \infty \).
Proof
Shrinking-box argument with \(\delta _t=t^{-2/5}\): the primary integral over \(\mathcal{B}_{\delta _t}\) converges to \(F(d,t)\) as \(t\to \infty \), and the residual is negligible for any fixed \(n\).