  In this paper, we study the distance problem in the setting of finite p-adic
rings. In odd dimensions, our results are essentially sharp. In even
dimensions, we clarify the conjecture and provide examples to support it.
Surprisingly, compared to the finite field case, in this setting, we are able
to provide a large family of sets such that the distance conjecture holds. By
developing new restriction type estimates associated to circles and orbits,
with a group theoretic argument, we will prove the $4/3$-parallel result in the
two dimensions. This answers a question raised by Alex Iosevich. In a more
general scenario, the existence/distribution of geometric/graph configurations
will be also considered in this paper. Our results present improvements and
extensions of recent results due to Ben Lichtin (2019, 2023). In comparison
with Lichtin's method, our approach is much simpler and flexible, which is also
one of the novelties in this paper.
