Analytic Capacity and the Subadditivity Problem

I studied for my doctorate at the University of Edinburgh from September 1970 to December 1973. I was awarded my PhD on the basis of my doctoral thesis, Analytic Capacity and the Subadditivity Problem, which is available for download from Edinburgh Research Archive.

It not easy to explain in layman’s terms what my thesis is about, but I will try, and I will start at the very beginning.

Real numbers

The real numbers are the ordinary numbers that the layman is familiar with: numbers like 2, −1.25, √ 3, π. The real numbers can be pictured geometrically as a horizontal line, known as the real line, extending infinitely far left and right, as shown in Figure 1.

Imaginary numbers

As well as the real numbers, mathematicians use so‐called imaginary numbers. The prototypical imaginary number is i, which has the property that i2 = −1. Real multiples of i, such as 2i and i 3, are also imaginary numbers. The imaginary numbers can be pictured geometrically as a vertical line, known as the imaginary line, extending infinitely far up and down the page, as shown in Figure 2. The adjective imaginary is unfortunate: imaginary numbers have just as strong a claim to existence as real numbers.

Complex numbers

When you add a real number and an imaginary number, you get a complex number. So for example −1 + 2i is a complex number, and π − i is a complex number. The complex numbers can be pictured geometrically as the points in a plane, known as the complex plane: a flat surface extending infinitely far in both dimensions, as shown in Figure 3.

Sets of complex numbers

A collection of elements with a given property is known as a set. On this page, I will be concerned exclusively with sets of complex numbers. I might talk, for example, of the set of all complex numbers whose distance from 0 is at most 1. That set is known as the closed unit disc. Characteristically, a set of complex numbers may be pictured as a geometrical shape in the complex plane, as for example the closed unit disc is pictured in grey in Figure 4.

Such visual representation takes us only so far however, since mathematicians are often concerned with messy sets, for example sets with infinitely many holes in them (affectionately known as Swiss cheeses), which our visual imagery may struggle with.

The union of two sets

If A and B are sets, the union of A and B, denoted A ∪ B, consists of all the points in A together with all the points in B. Figure 5 shows two closed discs A and B, each shaded in grey. The union of the two discs, A ∪ B, is represented by everything in grey.

How “big” is the closed unit disc?

If I define some set of complex numbers and ask how “big” it is, I may mean any of several things. How “big” is the closed unit disc? I might mean what is its cardinality: how many points are in it? Well, there are infinitely many points in it, but as there are infinitely many points in most sets that analysts are interested in, that doesn’t get us very far. I might mean what is its diameter, that is, how far apart can two points be if they are both in it or on its boundary? The answer to that is 2. I might mean what is its area, and the answer to that is π (because the area of a disc of radius r is πr2, and here the radius is 1).

Let’s stay with area for the moment. If I have two sets, A and B, and their areas are 5 and 3 respectively, then it follows that the area of their union A ∪ B is at most 5 + 3 = 8. The area of the union may be less than 8, if the two sets overlap, but it can’t be more than 8. If we denote the area of a set E by area(E), then we can say that:

area(A ∪ B) ≤ area(A) + area(B)

That is true of all sets A and B that are well enough behaved to have a well‐defined area, and we express that fact by saying that area is subadditive.

Now let’s look at diameter instead of area. The two discs in Figure 5, A and B, each have diameter 2. But the diameter of A ∪ B is 6: the leftmost point of A and the rightmost point of B are a distance 6 apart. And 6 > 2 + 2. Diameter is not subadditive.

Analytic capacity

We have seen that if I ask how “big” a set is, I might mean what is its cardinality, or what is its diameter, or what is its area. Or I might mean, what is its analytic capacity…

At this point, the mathematics suddenly becomes too difficult for the layman. Suffice it to say that analytic capacity is another interpretation of “bigness”, just as cardinality, diameter and area are interpretations of “bigness”. The analytic capacity of a set E is denoted γ(E). To give you the merest of flavours: the analytic capacity of a disc is its radius, and the analytic capacity of a straight line segment is a quarter of its length. The analytic capacity of a disc is the same as the analytic capacity of the hollow circle that forms its outer boundary: more generally, “holes” inside a set do not affect the analytic capacity.

The research question for my doctorate was: Given two sets A and B in the complex plane, can we always say that:

γ(A ∪ B) ≤ γ(A) + γ(B)?

In other words, is analytic capacity subadditive?

I don’t feel bad about not finding the answer in 3 years. Still nobody knows the answer, 45 years on. The question has proved to be a toughie. I did get several partial results, and these were good enough for my PhD.