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We had shown earlier that for a standard graded ring $R$ and a graded ideal$I$ in characteristic $p>0$, with $\ell(R/I) <\infty$, there exists a compactlysupported continuous function $f_{R, I}$ whose Riemann integral is the HKmultiplicity $e_{HK}(R, I)$. We explore further some other invariants, namelythe shape of the graph of $f_{R, {\bf m}}$ (where ${\bf m}$ is the gradedmaximal ideal of $R$) and the maximum support (denoted as $\alpha(R,I)$) of$f_{R, I}$.  In case $R$ is a domain of dimension $d\geq 2$, we prove that $(R, {\bf m})$is a regular ring if and only if $f_{R, {\bf m}}$ has a symmetry $f_{R, {\bfm}}(x) = f_{R, {\bf m}}(d-x)$, for all $x$.  If $R$ is strongly $F$-regular on the punctured spectrum then we prove thatthe $F$-threshold $c^I({\bf m})$ coincides with $\alpha(R,I)$.  As a consequence, if $R$ is a two dimensional domain and $I$ is generated byhomogeneous elements of the same degree, thene have (1) a formula for the$F$-threshold $c^I({\bf m})$ in terms of the minimum strong Harder-Narasimahanslope of the syzygy bundle and (2) a well defined notion of the $F$-threshold$c^I({\bf m})$ in characteristic $0$.  This characterisation readily computes $c^{I(n)}({\bf m})$, for the set ofall irreducible plane trinomials $k[x,y,z]/(h)$, where ${\bf m} = (x,y,z)$ and$I(n) = (x^n, y^n, z^n)$.

Hilbert-Kunz density functions and $F$-thresholds