Terence Tao

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 n, I multiply by the square root of 2, and I take the integer part of that.

So 10 by square root of 2 is like 14 point something, so 10 up to 14.

20 went 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-off errors, and sometimes when you add A plus B, this function doesn't quite give you the sum of the two individual outputs, but the sum plus or 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 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.

If you have these inverse theorems, it creates this dichotomy that either

the objects that you study either have no structure at all, or they are somehow related to something that is structured.

And in either case, you can make progress.

A good example of this is that there's this old theorem in mathematics called Szemeredi's theorem, proven in the 1970s.

It concerns trying to find a certain type of pattern in a set of numbers that the patterns have made progression.

Things like 3, 5, and 7, or 10, 15, and 20.

And Andrea Zamorelli 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.

So, for example, the odd numbers have a density of one-half,

and they contain rhythmic progressions of any length.

So in that case, it's obvious because the odd numbers are really, really structured.