A LeVeque-Type Inequality on the ring of $p$-adic integers

We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers ${\text{ZZ}}_p$ using techniques from Fourier analysis. The inequality is used to obtain an upper bound on the discrepancy of the sequence ${\alpha }_n=na+b$, where $a$ and $b$ are elements of ${\text{ZZ}}_p$. This is a $p$-adic analogue of the classical LeVeque inequality on the circle group $\text{RR}\mathrm{/}\text{ZZ}$.