Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 358 / 2 / iii

Letand, for $w\in \mathbb{C}$, form the finite rectangular matrixEvery entry is available from $\Lambda$. Let its singular values, padded and ordered as in the question, beFor fixed $n_2$, Parseval's identity gives convergence of the finite Gram matrices:as $n_1\to \infty$. Consequently, ifthenSingular values are Lipschitz under scalar shifts, so the subsequent limit $z_{n_2}\to z$ and part ii give

Define the finite-information arithmetic functionsFinite-matrix singular values can be obtained by arithmetic eigenvalue approximation, so these functions form the required ${\Delta }_4^A$ arithmetic tower using only $\Lambda$. The two inner limits recover the limiting singular values, while the outer limit is exactly the multiplicity formula from part ii: