We establish here some terminology, notation, and simple facts that will be used throughout the text.

Logarithms and exponentials

We write log x for the natural logarithm of x, and logbx for the logarithm of x to the base b.

We write ex for the usual exponential function, where e ≈ 2.71828 is the base of the natural logarithm. We may also write exp[x] instead of ex.

Sets and families

We use standard set-theoretic notation: ∅ denotes the empty set; x ∈ A means that x is an element, or member, of the set A; for two sets A, B, A ⊆ B means that A is a subset of B (with A possibly equal to B), and A ⊊ B means that A is a proper subset of B (i.e., A ⊆ B but A ≠ B). Further, A ∪ B denotes the union of A and B, A ∩ B the intersection of A and B, and A \ B the set of all elements of A that are not in B. If A is a set with a finite number of elements, then we write |A| for its size, or cardinality. We use standard notation for describing sets; for example, if we define the set S ≔ {−2,−1, 0, 1, 2}, then {x2 : x ∈ S} = {0, 1, 4} and {x ∈ S : x is even} = {−2, 0, 2}.