Let $p$ be a prime and $m \ge 1$.
Define the ($p$ residue of a residue) function
$\tag 1 \gamma_{(p,m)}: [1, pm] \to \{0,\dots,p-1\}$ $\tag 2 \quad \quad \quad \quad \quad \quad \quad \quad \quad \; n \mapsto [pm \pmod n] \pmod {p}$
Define
$\quad u_{(p,m)} = pm - \text{#}\bigl(\gamma_{(p,m)}^{-1}(0)\bigr)$
For $1 \le r \le p -1$ define
$\quad v_{(p,m,r)} = \frac{\text{#}\bigl(\gamma_{(p,m)}^{-1}(r)\bigr)}{u_{(p,m)}}$
Please prove that
$\tag 3 \lim_{m \to \infty} v_{(p,m,r)} = \frac{1}{p-1}$
While the proof is missing (or $\text{(3)}$ is refuted), allow me to make it a conjecture.
My Work
This came about from my investigations over here.