Say we are given a finite F set of unary and binary functions Z —> Z (like +, -, x, integer division, factorial) and a finite set J of allowable initial inputs (like {4}). I wonder if there may be mild conditions under which we can conclude that one or another of two possibilities holds: Maybe, the set of all obtainable numbers has a) density = 0 in Z, or else b) density = 1 in Z. (Or some other dramatic dichotomy if not that one.) The mild conditions might avoid the situations where all numbers obtained avoid some remainder mod p for some prime p. —Dan