Sum-product, unit distances, and number fields (opens in new tab)

(erdosproblems.com)

67 pointsrobinhouston24d ago19 comments

19 comments

9 comments · 3 top-level

amelius20d ago· 6 in thread

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.

bustermellotron20d ago

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

amelius20d ago

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?

7 more replies

LPisGood20d ago

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.

arijun20d ago

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.

Ar-Curunir20d ago

That is just exponentiation, eg squaring. They’re just putting symbols instead of numbers there.

1 more reply

pfdietz20d 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.

fithisux20d ago

Unfortunately it does not render correctly in my browser. If you could create a Latex or PDF would be very good.

j / k navigate · click thread line to collapse