Describe set of integers $x$ such that the $p$-adic norm of $(x-a)$ is smaller than $p^{-e}$

Question

How to describe set of integers $x$ such that the $p$-adic norm of $(x-a)$ is smaller than $p^{-e}$?

i.e. How can I use the language of congruence to describe the set of integers $x$ such that $|x-a|p< p^{-e}$?

Answer 1

For a given integer $a$ and prime number $p$, the $p$-adic norm of $x-a$ is smaller than $p^{-e}$ if and only if $v_p(x-a)>e$. That is to say, if and only if $x-a$ is divisible by a power of $p$ greater than $e$, i.e. $x-a=p^{e+1}b$ for some integer $b$. In this way we find that your set of integers is precisely the set $$\{a+bp^{e+1}:\ b\in\Bbb{Z}\}.$$ In terms of congruences this can be written as $$\{x\in\Bbb{Z}:\ x\equiv a\pmod{p^{e+1}}\}.$$