Sum-product, unit distances, and number fields
erdosproblems.com65 points by robinhouston 5 days ago
65 points by robinhouston 5 days ago
The interesting thing here is he's explaining how the new idea that the AI came up with for the unit distance problem was then applied by a group of human mathematicians to refute the sum-product conjecture. One result is not a consequence of the other; it's a tool invented for the first that could be reapplied, in a modified form, for the second.
Some of those involved in the sum-product result helped digest the AI result, simplifying it and extracting the useful insight.
Some have warned about AI producing inscrutable results, but just the opposite seems to have happened here.
Unfortunately it does not render correctly in my browser. If you could create a Latex or PDF would be very good.
Stopped reading when they used |·| without explaining its meaning.
> In this blog post I will give my personal view on the recent counterexamples to the unit distance conjecture and sum-product conjecture over the reals (see [90] and [52] respectively). My goal is to sketch the constructions and try and give some intuition as to where they came from and why they work. *My main target audience is the me-of-a-month-ago*
Thomas Bloom is a world-class mathematician working at the frontier of his field. He explicitly did not write this for you, and your stopping reading it is fine.
Size of a finite set. It’s common notation in this field.
Ok, I guess I'm always confused why |S| on a set doesn't take the values into account, whereas |x| on a vector does take its values into account; how can mathematicians of all people be so inconsistent?
Think of it this way, how would you turn a set into a vector in the first place? We solve this in programming a lot, for example, the "one-hot" encoding for neural networks. Here, a set turns into a vector that has a zero for every item that isn't in the set, a one for every item that is.
Now, there are a lot of things that |v| for a vector can mean. In the L1 distance you just add up the absolute value of each dimension. You could argue that that's a simpler sort of |v| than L2.
And there you go! |S| on a set actually means exactly the same thing as |x| on a vector, if you interpret sets as vectors in the right way.
There is a joke saying "a mathematician says X, writes Y on the board and means Z". The really amusing(?) thing is that other mathematicians still (sort of) perfectly understands Z. Once you have enough experience you fill in the blanks automatically.
Math exposition is tricky: too few details and you're just floating in the sky, too many details and the audience loses sight of the forest for all the trees. You can go (more or less) all formal, but it's a pain for the writer and a pain for the experienced reader.
If it's any consolation, the punchline to the joke is that it often is small/big lie: the other mathematicians reads "Y" and goes WTF!? And then 1 minute, 1 hour, 1 day, or one week later says "aaah, that's what he/she meant! I guess it was 'obvious' all along". :-)
Even Terry Tao struggled at times: "When I was a graduate student in Princeton, Tom Wolff came and gave a course on recent progress on the restriction and Kakeya conjectures, starting from the breakthrough work of Jean Bourgain in a now famous 1991 paper in Geom. Func. Anal.. I struggled with that paper for many months; it was by far the most difficult paper I had to read as a graduate student, as Jean would focus on the most essential components of an argument, treating more secondary details (such as rigorously formalising the uncertainty principle) in very brief sentences."
More details at
https://terrytao.wordpress.com/2018/12/29/jean-bourgain/
(In particular, see the "???" in the Tao's annotated copy of Bourgain's paper.)
Mathematicians use inconsistent notations all the time. Symbols meaning slightly different things based on the type of the arguments are among the benign cases.
For the present case, see https://en.wikipedia.org/wiki/Vertical_bar#Mathematics.
I dunno I think it makes sense. For a vector x, the length |x| says something about its size relative to other vectors. For a set S, the cardinality |S| says something about its size relative to other sets.
The vector is always defined in a vector field which has a given dimension, and usually the dimension isn't that interesting. Typically it's either the same between the vectors you consider, or the vectors have one of a few fixed number of dimensions. Meanwhile the length of vectors is an interesting quantity.
For sets, since the values can be anything, nothing or everything in between, you can't really define many interesting functions or operations that work on the elements of sets in general. Meanwhile, the number of elements in a set is an interesting quantity.
Anyway, just my take, though I never did take much math.
|x| on a vector is not well-typed. The notation for norms of vectors is ||x||, namely 2 bars on each side.
There still is a naming collision between |X| (cardinality of a set) and |x| (absolute value of a scalar). Sometimes this happens. It generally is still unambiguous though, as mathematicians tend to use different segments of the alphabet for different purposes, and also additionally tend to capitalize sets, and leave scalars lower case.
I stopped reading earlier, when they used superscript without explaining its meaning. Its clearly meant for someone with more domain expertise than me, with my hazy recollections of college math.
That is just exponentiation, eg squaring. They’re just putting symbols instead of numbers there.