On certain dense sets of rationals, simultaneous Schröder and Böttcher equations, and characterizations of functions (Q1861180)

Two integers \(a\) and \(b\), each greater than 1, are power-disjoint, if \(a^m\neq b^n\) for all pairs \((m,n)\) of positive integers. Theorem 1. Let \(a\) and \(b\) be two power-disjoint integers. Then the set of rational numbers of the form \(a^{-m} b^n\), where \(m\) and \(n\) (independently) run through the nonnegative integers, is dense in \(\mathbb R^+\). This theorem is used to study the following pairs of functional equations \[ f(ax)= af(x),\quad f(bx)= bf(x), \] \[ f(x^a)= af(x),\quad f(x^b)= bf(x), \] \[ f(ax)= (f(x))^a,\quad f(bx)= (f(x))^b, \] \[ f(x^a)= (f(x))^a,\quad f(x^b)= (f(x))^b. \]