We worked a 60 hour week at Bomber Command, so we were pretty heavily engaged - but still there were some times left in the evenings when I did mathematics just to keep myself sane. And then I did my first actual, I would say, serious contribution to pure mathematics, which was the alpha-beta theorem, which was a theorem about the sum sequence of sequences of integers. If you take a couple of sequences of integers, then you can make another sequence by summing them in pairs, and you add together an A and a B and you put together all the A + B, it makes a new sequence, C. And the theorem says the density of C is at least as great as the sum of the densities of A and B. And that theorem was proved by Henry Mann in 1942, during the war. He was at Ohio State, and he's still around, I think. He was a very young student at the time. It was a very difficult and beautiful piece of work.
[Q] And you read that paper while you were at Cambridge?
Yes. So I read that paper while I was at Cambridge, and it was obvious that it would be nice to extend that to more than two sequences, and it wasn't obvious that one could do that. And so I set myself that as a problem and I solved it, so that the theorem I proved was the extension of Mann's theorem to more than two sequences. So if you have several sequences and you add one term from each, then the sum has a density which is at least the sum of the densities. And that was one of the most satisfying things I ever did. It's a really beautiful piece of work. The proof is elegant. It's not at all obvious and it's just a sort of demonstration of how beautiful mathematics can be, with just the most simple and elementary materials, nothing but integers, all finite arguments, and of course a bit of a Besicovitch flavour to it because it involves a hierarchical construction very much in the Besicovitch style, although it was quite different from the problems that Besicovitch was interested in. So that I actually wrote during the war and it was published in the Journal of the London Mathematical Society.
[Q] During the war?
No, I think only afterwards, but anyway it's my first substantial published work. And it was very helpful because it got me a Trinity College Fellowship.
