Extremal permutations of $\mathbb{Z}_n$ with many good slopes and applications to quantum Latin squares

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 prove the sharp upper bound $φ(n)$ on the number of good slopes, where $φ$ is Euler's totient function. We classify and count the equality cases for prime powers and products of two distinct odd primes. Non-affine equality cases exist for every composite $n\ge6$. We also prove the sharp bound $p_0-1$ for the size of a set of good slopes with invertible differences between distinct elements, where $p_0$ is the least prime divisor of $n$. Finally, we establish gap results below $φ(n)$ and apply our bounds to a construction of quantum Latin squares due to Ball and Simoens.