The Hilbert-Kunz function of some quadratic quotients of the Rees algebra

Given a commutative local ring \((R,\mathfrak m)\) and an ideal \(I\) of \(R\), a family of quotients of the Rees algebra \(R[It]\) has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When \(R\) is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either \(I\) is \(\mathfrak{m}\)-primary or \(R\) is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored.