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CelticBard
You might want to change the title of the post. The article's title is very interesting but just saying "Quant Magazie" in the title isn't interesting
coolandsmartrr
Looks like this submission links to the article "From Systems in Motion, Infinite Patterns Appear", which is a title that's far more interesting than the name of its publication
dang
Fixed now. Thanks!
whearyou
Can someone explain this for the cretins in the room?
mauriciolange
Still not a sufficient explanation but I pretty much like the headline:

Mathematicians are finding inevitable structures in sufficiently large sets of integers.

_a_a_a_
The word structure is commonly used but never seems to be defined.
somrand0
you need to study lots of higher algebra to get a sense of what they mean...

all I can tell you (because it's all I know) is that "the structure" is preserved by transformations.

and what the subject focuses on and studies is the transformations, more than 'the structures'.

so I suppose they don't precisely and explicitly know what 'the structure' is; but the incredible thing is how this does NOT matter.

as far as I've figured out so far, the point is that it gets preserved across transformations, and that they can inter-relate it across different mathematical objects.

rightbyte
Won't a sufficiently large set contain any pattern?
brookst
Assuming a random distribution in the set and any finite pattern, I think so?
badrabbit
Why? E.g. numbers that start with 1 are more commonly seen in random sets than other numbers. Things might repeat forever in a pattern
tromp
In 2018, some mathematicians proved that any non-negative integer subset with a positive limit density (for some eps>0 and all large enough n, the subset contains at least eps*n numbers below n) contains a sumset, i.e. a set C that can be written as the sum of two infinite sets C = A + B (the set of a+b where a in A and b in B). More recently, they extended this to sums of more than two infinite sets.
_a_a_a_
I'm afraid that does not illuminate at all.
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brookst
Can someone explain this comment for the sub-cretins in the room?
Someone
Let’s start with finite sets, and in reverse: given the two sets A and B, define set

  C = {a + b | a ∈ a, b ∈ B}
For example if

  A = {1, 3, 7}
and

  B = {1, 4, 8}
, we have

  C = {1 + 1, 1 + 4, 1 + 8,
       3 + 1, 3 + 4, 3 + 8,
       7 + 1, 7 + 4, 7 + 8}
    = {2, 5, 9, 4, 7, 11, 8, 11, 15}
    = {2, 4, 5, 7, 8, 9, 11, 15}
That’s easy. Now, if you didn’t know that, but were given set C, can you find two sets A and B of size 3 that would produce C under that logic?

An easy (but cumbersome) way to answer that is: produce all sets of 3 positive integers less than the largest number in C (15 in the example), compute the respective C’s and check whether that’s equal to the targeted C.

This problem is about targeting infinite sets of integers for A, B, and C, with an extra twist: for what infinite sets of integers C can you find matching infinitely sized sets A and B that both are subsets of C?

Since all of these sets are infinite, the easy but cumbersome method won’t work; neither set has a largest element. In addition, the result isn’t about a specific set of integers.

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