Good slopes, congruence preservation, and gaps over residue rings

Abstract

For $n\ge2$, a good slope of a permutation $f$ of $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$ is an element $m\in\mathbb{Z}_n$ for which $x\mapsto f(x)+mx$ is a permutation. We call $f$ congruence-preserving if it preserves congruences modulo every divisor of $n$. For odd prime powers $n$, we classify all permutations with more than $n/2$ good slopes. Among permutations that are not congruence-preserving, we determine the maximum number of good slopes for $n=3^e$ with $e\ge2$, and for $n=25$, $125$ and $625$. Every permutation of $\mathbb{Z}_n$ has at most $φ(n)$ good slopes, where $φ$ is Euler's totient function. For several families of moduli with at least two distinct prime divisors, we prove a sharp upper bound for the number of good slopes of the permutations with fewer than $φ(n)$ good slopes. These families include $9q$ and $3q^e$ for every prime $q\ge5$ and every $e\ge1$. Together with a companion paper, this settles the bound for every odd composite modulus below $105$.