From the results of Section 1.2 it is clear that sets such as ∅, {∅}, {∅, {∅}}, {{∅}}, and so forth exist. Using the axiom of infinity we can also conclude that we can build similar infinite sets. But how do we construct complicated sets, such as the sets of natural and real numbers, if the only tools we have to build them are the empty set ∅ and “braces” {·}? We will solve this problem by identifying the aforementioned objects with some specific sets.

Briefly, we will identify the number 0 with the empty set ∅, and the other natural numbers will be constructed by induction, identifying n with {0, …, n − 1}. The existence of the set ℕ of all natural numbers is guaranteed by the infinity axiom. The real numbers from the interval [0,1] will be identified with the set of functions {0,1}ℕ, where an infinite sequence a: ℕ → {0,1} is identified with the binary expansion of a number, that is, with Σn∈ℕa(n)/2n+1. The details of these constructions are described in the rest of this chapter.

Natural numbers

In this section we will find a set that represents the set ℕ of natural numbers in our set-theoretic universe. For this, we need to find for each natural number n a set that represents it. Moreover, we will have to show that the class of all such defined natural numbers forms a set.

Why axiomatic set theory?

Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its basics: elementary operations on sets, fundamental facts about functions, and, in some cases, rudimentary elements of cardinal arithmetic. This basic part of set theory is very intuitive and can be developed using only our “good” intuition for what sets are. The theory of sets developed in that way is called “naive” set theory, as opposed to “axiomatic” set theory, where all properties of sets are deduced from a fixed set of axioms.

Clearly the “naive” approach is very appealing. It allows us to prove a lot of facts on sets in a quick and convincing way. Also, this was the way the first mathematicians studied sets, including Georg Cantor, a “father of set theory.” However, modern set theory departed from the “paradise” of this simple-minded approach, replacing it with “axiomatic set theory,” the highly structured form of set theory. What was the reason for such a replacement?

Intuitively, a set is any collection of all elements that satisfy a certain given property. Thus, the following axiom schema of comprehension, due to Frege (1893), seems to be very intuitive.

A more accurate title would be: Introduction to some aspects of the analytic theory of automorphic forms on SL2(ℝ) and the upper half-plane X. Originally, automorphic forms were holomorphic or meromorphic functions on X satisfying certain conditions with respect to a discrete group Γ of automorphisms of X, usually with fundamental domain of finite (hyperbolic) area. Later on, H. Maass – and then A. Selberg and W. Roelcke – dropped the assumption of holomorphicity, requiring instead that the functions under consideration be eigenfunctions of the Laplace–Beltrami operator. In the 1950s it was realized (in more general cases) – initially by I. M. Gelfand and S. V. Fomin, and then by Harish-Chandra – that the automorphic forms (holomorphic or not) could be equivalently viewed as functions on Γ\SL2(ℝ) satisfying certain conditions familiar in the theory of infinite dimensional representations of semisimple Lie groups. This led to a new outlook, where the Laplace–Beltrami operator is replaced by the Casimir operator and the theory of automorphic forms becomes closely related to harmonic analysis on Γ\SL2(ℝ). This is the point of view adopted in this presentation. However, in order to limit the prerequisites, no knowledge of representation theory is assumed until the last sections, a main purpose of which is precisely to make this connection explicit. A fundamental role is played throughout by a theorem stating that a function on SL2(ℝ) satisfying certain assumptions (Z-finite and K-finite) is fixed under convolution by some smooth function with arbitrarily small compact support around the identity element (2.14).

Strange subsets of ℝn and the diagonalization argument

In this section we will illustrate some typical constructions by transfinite induction. We will do so by constructing recursively some subsets of ℝn with strange geometric properties. The choice of geometric descriptions of these sets is not completely arbitrary here – we still have not developed the basic facts concerning “nice” subsets of ℝn that are necessary for most of our applications. This will be done in the remaining sections of this chapter.

To state the next theorem, we will need the following notation. For a subset A of the plane ℝ2 the horizontal section of A generated by y ∈ ℝ (or, more precisely, its projection onto the first coordinate) will be denoted by Ay and defined as Ay = {x ∈ ℝ: 〈x,y〉 ∈ A}. The vertical section of A generated by x ∈ ℝ is defined by Ax = {y ∈ ℝ: 〈x,y〉 ∈ A}.

We will start with the following example.

Theorem 6.1.1There exists a subset A of the plane with every horizontal section Ay being dense in ℝ and with every vertical section Ax having precisely one element.

Proof We will define the desired set by induction. To do so, we will first reduce our problem to the form that is most appropriate for a recursive construction.

The requirement that every horizontal section of A is dense in ℝ tells us that A is “reasonably big.”

The spectral decomposition ends this description of some basic material on the analytic theory of automorphic forms on SL2(ℝ). This is, however, only the starting point of the theory. We now add some comments on other developments, mainly for orientation and to indicate some literature for further study, without aiming at completeness.

Let A(s, m) be the space of automorphic forms of right K-type m that are eigenfunctions of C with eigenvalue (s2 - 1)/2. By definition, this space is the orthogonal direct sum of the subspace °A(s, m) of cusp forms and of its orthogonal complement A1(s, m). In Section 12, we obtained some information on the latter: its dimension is equal to the number of cusps or neat cusps; it is generated by Eisenstein series E(r, m) holomorphic at s and suitable limits of Eisenstein series. Although this description is not quite exhaustive, depending notably on the poles of Eisenstein series, it is quite substantial, so that the main remaining issue is the determination of the cusp forms. Note that, in the cocompact case, A1(s, m) = {0} and so nothing has been achieved toward the description of A(s, m).

By 16.2, the cuspidal spectrum °L2(Γ\G) is a Hilbert direct sum of irreducible unitary G-modules π ∈ Ĝ with finite multiplicities, say m(π, Γ). We know the K-types of all irreducible unitary representations of G (see §15), and they all have multiplicity 1 (a very special property of SL2(ℝ)).

Here again, we let H = L2(Γ\G), δH = °L2(Γ\G), and V be the orthogonal complement of °H in H.

The space H is the Hilbert direct sum of the subspace Hm (m ∈ ℤ), so we already have a spectral decomposition of H with respect to C. The list in 15.8 shows that there are at most two inequivalent irreducible unitary representations of G with the same Casimir operator, so there is in principle not much difference between the decompositions with respect to C and to G. But we want to express the latter in terms of representations.

Lemma. Let (π, E) be a unitary representation of G. Assume the existence of a Dirac sequence {αn} (n ∈ ℕ) with compact operators π(αn). Then E is a Hilbert sum of irreducible representations, with finite multiplicities.

Proof. This is rather elementary and well-known. For the convenience of the reader we reproduce one proof, following 5.8 and 5.9 in [7].

(a) We show first that a closed G-invariant subspace W ≠ 0 contains a closed G-invariant subspace that is minimal among nonzero closed G-invariant subspaces. By 13.1 and 14.2, there exists a j such that π(αj)|w ≠ 0. Let c be a nonzero eigenvalue of π(αj) in W, and let M ≠ 0 be the corresponding (finite dimensional) eigenspace in W.