Scheme: University Research Fellowship
Organisation: University of Oxford
Dates: Oct 2016-Sep 2019
Summary: This project summary is not available for publication.
Dates: Oct 2011-Sep 2016
Summary: Everyone encounters the whole numbers in their daily lives, and it is the job of some mathematicians to try to understand the whole numbers. It might seem that this would just involve ‘working out bigger and bigger equations’ but this is a bit like imagining the job of a cook is to make a bigger and bigger sponge. Sure, they could, but a really good cook understands his ingredients well enough that he can make a whole range of really great cakes.
Suppose that you are allowed to choose 1% of the whole numbers less than 1,000,000 in any way you like. Can I guarantee that however you’ve done it there are three consecutive numbers? Clearly not as you could just pick 1,101,201,301,.... However, this collection still has three numbers that are sort of ‘in a row’, in the sense that the difference between 101 and 201 is the same as between 201 and 301. When we have three numbers where the difference between the first two is equal to the difference between the second two it is said to be an arithmetic progression, and it turns out however you picked the original 1% of numbers you will have been forced to pick an arithmetic progression. This is called Roth’s theorem.
This may just seem like a cute fact, but what we are really interested in, like all scientists, is why it is true. What is it about having chosen 1% of the numbers that forces these arithmetic progressions to exists? Could we choose a smaller percentage? Can we choose the same percentage of a smaller number? In a sense answering these questions is one of the aims of this project.
What is particularly intriguing about the problem above is that while it seems to have a lot to do with numbers it is quite possible that this is an illusion. It actually turns out that in a certain sense the best way we know to pick lots of numbers without arithmetic progressions makes them look like a high dimensional sphere – the problem seems to be geometric rather than number theoretic.