On the fixed points of the map $x \mapsto x^x$ modulo a prime, II

We study number theoretic properties of the map $x \mapsto x^{x} \mod{p}$, where $x \in \{1,2,\ldots,p-1\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map only has the trivial fixed point $x=1$. A key technical result, possibly of independent interest, is the existence of subsets $\mathscr{N}_{q} \subset \{2,3,\ldots,q-1\}$ such that almost all $k$-tuples of distinct integers $n_{1}, n_{2},\ldots,n_{k} \in \mathscr{N}_q$ are multiplicatively independent (if $k$ is not too large), and $|\mathscr{N}_q| = q \cdot (1+o(1))$ as $q \to \infty$. For $q$ a large prime, this is used to show that the number of solutions to a certain large and sparse system of $\mathbb{F}_q$-linear forms $\{ \mathscr{L}_{n} \}_{n=2}^{q-1}$ "behaves randomly" in the sense that $|\{ \mathbf{v} \in \mathbb{F}_{q}^{d} : \mathscr{L}_{n}(\mathbf{v}) =1, n = 2,3, \ldots, q-1 \}| \sim q^{d}(1-1/q)^{q} \sim q^{d}/e$. (Here $d=\pi(q-1)$ and the coefficents of $\mathscr{L}_{n}$ are given by the exponents in the prime power factorization of $n$.)