HK multiplicity, $F$-threshold and the Paley-Wiener theorem

For a given algebraically closed field $k$ of characteristic $p>0$ we consider the set ${\mathcal C}_k$, of graded isomorphism classes of {\em standard graded pairs} $(R, I)$, where $R$ is a standard graded ring over the field and $I$ is a graded ideal of finite colength. Here we give a ring homomorphism $\Pi:\Z[{\mathcal C}_k] \longrightarrow H(\C)[X]$, where $H(\C)$ denotes the ring of entire functions. The related entire function and the homomorphism $\Pi$ keep track of the two positive characteristic invariants, $e_{HK}(R, I)$ and $c^I({\bf m})$ of the ring: (1) composing the map $\Pi$ with the evaluation map at $z=0$ gives a ring homomorphism $\Pi_e:\Z[{\mathcal C}_k] \longrightarrow \R[X]$ which sends $$(R,I) \to e_{HK}(R^0, IR^0)+ e_{HK}(R^1, IR^1)X+\cdots + e_{HK}(R^d, IR^d)X^d,$$ where $R^i$ is the union of $i$ dimensional components of $R$ and $e_{HK}(R^i, IR^i)$ is the HK multiplicity of the pair $(R^i, IR^i)$, and in particular the top coefficient is $e_{HK}(R, I)$. (2) If, in addition, $R$ is a two dimensional ring or $\mbox {Proj~R}$ is strongly $F$-regular, then the Fourier transform ${\widehat f}_{R, I}$ belongs to the Paley-Wiener class of the real number, namely the $F$-threshold $c^I_{\bf m}(R)$ of the maximal ideal ${\bf m}$.