Terence Tao: preference

Transcript context

…Yeah. This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness, that most objects that you can generate in mathematics are random. They look like random, like the digital supply, well, we believe is a good example. But there’s a very small number of things that have patterns. But now, you can prove something has a pattern by just constructing… If something has a simple pattern and you have a proof that it does something like repeat itself every so often, you can do that and you can prove that… For example, you can prove that most sequences of digits have no pattern. So, if you just pick digits randomly, there’s something called low-large numbers. It tells you you’re going to get as many ones as twos in the long run. But we have a lot fewer tools to… If I give you a specific pattern like the digits of pi, how can I show that this doesn’t have some weird pattern to it? Some other work that I spent a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured. So, some functions are what’s called additive. If you have a function of natural numbers of the natural numbers, so maybe two maps to four, three maps to six and so forth, some functions are what’s called additive, which means that if you add two inputs together, the output gets added as well. For example, a multiply by constant. If you multiply a number by 10… If you multiply A plus B by 10, that’s the same as multiplying A by 10 and B by 10, and then adding them together. So, some functions are additive, some functions are kind of additive but not completely additive. So, for example, if I take a number, and I multiply by the square of two and I take the integer part of that, so 10 by square route of two is like 14 point something, so 10 up to 14, 20 or up to 28. So, in that case, additivity is true then, so 10 plus 10 is 20 and 14 plus 14 is 28. But because of this rounding, sometimes there’s round-up errors, and sometimes when you add A plus A, this function doesn’t quite give you the sum of the two individual outputs, but the sum plus/minus one. So, it’s almost additive, but not quite additive. So, there’s a lot of useful results in mathematics, and I’ve worked a lot on developing things like this, to the effect that if a function exhibits some structure like this, then it’s basically there’s a reason for why it’s true. And the reason is because there’s some other nearby function, which is actually completely structured, which is explaining this sort of partial pattern that you have. And so if you have these inverse theorems, it creates this dichotomy that either the objects that you study are either have no structure at all or they are somehow related to something kind of structured. And in either way, in either case, you can make progress. A good example of this is that there’s this old theorem in mathematics- re at all or they are somehow related to something kind of structured. And in either way, in either case, you can make progress. A good example of this is that there’s this old theorem in mathematics- A good example of this is that there’s this old theorem in mathematics called Szemerédi’s Theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers, the patterns of arithmetic progression. Things like three, five, and seven or 10, 15 and 20, and Szemerédi, Endre Szemerédi proved that any set of numbers that are sufficiently big, what’s called positive density, has arithmetic progressions in it of any length you wish. For example, the odd numbers have a density of one half, and they contain arithmetic progressions of any length. So in that case, it’s obvious, because the odd numbers are really, really structured. I can just take 11, 13, 15, 17, I can easily find arithmetic progressions in that set, but Szemerédi’s theorem also applies to random sets. If I take a set of odd numbers and I flip a coin for each number, and I only keep the numbers for which I got a heads… So I just flip coins, I just randomly take out half the numbers, I keep one half. That’s a set that has no patterns at all, but just from random fluctuations, you will still get a lot of arithmetic progressions in that set. Can you prove that there’s arithmetic progressions of arbitrary length within a random-…

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