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 $\mathrm{\Pi }:\mathbb{Z}[{\mathcal{C}}_k]⟶H(\text{C})[X]$, where $H(\text{C})$ denotes the ring of entire functions. The related entire function and the homomorphism $\mathrm{\Pi }$ keep track of the two positive characteristic invariants, $e_{HK}(R,I)$ and $c^I(\mathbf{m})$ of the ring: (1) composing the map $\mathrm{\Pi }$ with the evaluation map at $z=0$ gives a ring homomorphism ${\mathrm{\Pi }}_e:\mathbb{Z}[{\mathcal{C}}_k]⟶\mathbb{R}[X]$ which sends $$(R,I)\to e_{HK}(R^0,I{R}^0)+e_{HK}(R^1,I{R}^1)X+\cdots +e_{HK}(R^d,I{R}^d)X^d,$$ where $R^i$ is the union of $i$ dimensional components of $R$ and $e_{HK}(R^i,I{R}^i)$ is the HK multiplicity of the pair $(R^i,I{R}^i)$, and in particular the top coefficient is $e_{HK}(R,I)$. (2) If, in addition, $R$ is a two dimensional ring or $\text{mbox}ProjR$ 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 $c_{\mathbf{m}}^I(R)$ of the maximal ideal $\mathbf{m}$.