Numbers having $m$ small $m$th roots mod $p$

Abstract:

Here are two typical results about the numbers mentioned in the title: If p is a prime such that $p \equiv 1 \pmod 6$ and $p > 67$, then there are exactly six numbers $\bmod \;p$, each of which has six sixth roots less than $2\sqrt {3p}$ in absolute value. If p is a prime such that $p \equiv 1 \pmod 8$, then there is at least one number $\bmod \;p$ which has eight eighth roots less than ${p^{3/4}}$ in absolute value.