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

## TL;DR

This paper establishes a LeVeque-type inequality for sequences in the ring of p-adic integers, providing bounds on their discrepancy using Fourier analysis, extending classical results to a p-adic setting.

## Contribution

It introduces a p-adic analogue of the LeVeque inequality, applying Fourier analysis to derive discrepancy bounds on sequences in Zp.

## Key findings

- Derived a discrepancy inequality for p-adic sequences
- Applied the inequality to linear sequences in Zp
- Extended classical discrepancy results to p-adic integers

## Abstract

We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers $\ZZ_p$ 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 $\ZZ_p$. This is a $p$-adic analogue of the classical LeVeque inequality on the circle group $\RR/\ZZ$.