Integral automorphisms of affine spaces over finite fields

Let $GF(q)$ denote the finite field with $q$ elements, and let $S$ denote the set of all square elements of $GF(q)$. The norm $N$ of a point $x=(x_1,\ldots,x_n)$ of the affine space $AG(n,q)$ is defined by $N(x) = x_1^2+\cdots+x_n^2,$ and two points $x$ and $y$ are said to be at integral distance if $N(x-y)$ is in $S$. An integral  automorphism of $AG(n,q)$ is a permutation of the point set which  preserves integral distances. In this talk I present some recent results on integral automorphisms. The talk is based on a joint work with Klavdija Kutnar, J\’anos Ruff and Tam\’as Sz\H onyi.