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Sum-product, unit distances, and number fields

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Re: Sum-product, unit distances, and number fields

#11

Post by kurlberg »

kurlberg wrote: post #9
amelius wrote: post #4

Earlier quoted context omitted.

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?

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 g…

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.)

Re: Sum-product, unit distances, and number fields

#12

Post by Gregaros »

amelius wrote: post #2

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.

Re: Sum-product, unit distances, and number fields

#13

Post by LPisGood »

bustermellotron wrote: post #3
amelius wrote: post #2

Stopped reading when they used |·| without explaining its meaning.

Size of a finite set. It’s common notation in this field.

This is extremely common notation, and it’s common across STEM fields. If you’ve done any discrete math, algorithms analysis, etc. this should be familiar to you.

Re: Sum-product, unit distances, and number fields

#14

Post by lacker »

amelius wrote: post #4
bustermellotron wrote: post #3

Earlier quoted context omitted.

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.

Re: Sum-product, unit distances, and number fields

#15

Post by Ar-Curunir »

arijun wrote: post #8
amelius wrote: post #2

Stopped reading when they used |·| without explaining its meaning.

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.

Re: Sum-product, unit distances, and number fields

#16

Post by mswphd »

amelius wrote: post #4
bustermellotron wrote: post #3

Earlier quoted context omitted.

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?

|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.

Re: Sum-product, unit distances, and number fields

#17

Post by pfdietz »

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.

Re: Sum-product, unit distances, and number fields

#19

Post by jjgreen »

mswphd wrote: post #16
amelius wrote: post #4

Earlier quoted context omitted.

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?

|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…

Sorry to be that guy, but actually ‖x‖

Re: Sum-product, unit distances, and number fields

#20

Post by arijun »

Ar-Curunir wrote: post #15
arijun wrote: post #8

Earlier quoted context omitted.

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.

But then they say A + A <= A^2, wouldn’t that be false for a = 1?
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