"Sumsets", and, "Borel vis-a-vis Combinatorics"

[Ross Finlayson]

"But Ruzsa’s theorem required the subgroup to be enormous.
Marton’s insight was to posit that rather than being contained
in one giant subgroup, A could be contained in a polynomial
number of cosets of a subgroup that is no bigger than the original set A."

-- https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/

(The Quanta Magazine's are usually puff-pieces, but often about
interesting things.)

Here for example about the integers, here and generally natural integers,
there's a usual consideration about sequences of 0's and 1's, representing
subsets of the integers, with all 0's being empty and all 1's being all.
So, Ruzsa's theorem is about all, but would be, "larger" the set, then
that the researchers figured out a way to "attain" to it, then for something
like, "Falting's purity", whether the, "almost everywhere", as it were,
is not a hypocritical assignment.

https://arxiv.org/abs/2306.13403

Hmm, more about "doubling", here as about usually "doubling spaces"
and "doubling measures", about Ramsey theory, about the usual notion
of *-distribution or equi-distribution, of the 0/1 sequences, and that
the _language_ of the sequences and _space_, of the sequences, are
not the same thing, for Cantor space and Square Cantor space,
and, "Borel versus Combinatorics".

I.e., otherwise people could point at Ruzsa's theorem, and, in
a sense, point out a corresponding incompatibility, and show
counterexamples to these above theorems.

I.e., the rigorous formalist would sort of note that what really
needs be is a theory where Vitali sets make for Dirichlet and Poincare
their real analytical character, and that Banach-Tarski has quite different
arrivals of the geometric (the space) and algebraic (the words),
about the entire equi-interpretability, of otherwise these sorts results.

[Ross Finlayson]

Post by Ross Finlayson
"But Ruzsa’s theorem required the subgroup to be enormous.
Marton’s insight was to posit that rather than being contained
in one giant subgroup, A could be contained in a polynomial
number of cosets of a subgroup that is no bigger than the original set A."
-- https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/
(The Quanta Magazine's are usually puff-pieces, but often about
interesting things.)
Here for example about the integers, here and generally natural integers,
there's a usual consideration about sequences of 0's and 1's, representing
subsets of the integers, with all 0's being empty and all 1's being all.
So, Ruzsa's theorem is about all, but would be, "larger" the set, then
that the researchers figured out a way to "attain" to it, then for something
like, "Falting's purity", whether the, "almost everywhere", as it were,
is not a hypocritical assignment.
https://arxiv.org/abs/2306.13403
Hmm, more about "doubling", here as about usually "doubling spaces"
and "doubling measures", about Ramsey theory, about the usual notion
of *-distribution or equi-distribution, of the 0/1 sequences, and that
the _language_ of the sequences and _space_, of the sequences, are
not the same thing, for Cantor space and Square Cantor space,
and, "Borel versus Combinatorics".
I.e., otherwise people could point at Ruzsa's theorem, and, in
a sense, point out a corresponding incompatibility, and show
counterexamples to these above theorems.
I.e., the rigorous formalist would sort of note that what really
needs be is a theory where Vitali sets make for Dirichlet and Poincare
their real analytical character, and that Banach-Tarski has quite different
arrivals of the geometric (the space) and algebraic (the words),
about the entire equi-interpretability, of otherwise these sorts results.

Doubling, ..., also about quasi-invariant measure theory and for physics,
quite about the continuous and discrete, and they meet in the middle,
doubling, ....

How many Fields' medalists do you see?

[Ross Finlayson]

Post by Ross Finlayson

Post by Ross Finlayson
"But Ruzsa’s theorem required the subgroup to be enormous.
Marton’s insight was to posit that rather than being contained
in one giant subgroup, A could be contained in a polynomial
number of cosets of a subgroup that is no bigger than the original set A."
-- https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/
(The Quanta Magazine's are usually puff-pieces, but often about
interesting things.)
Here for example about the integers, here and generally natural integers,
there's a usual consideration about sequences of 0's and 1's, representing
subsets of the integers, with all 0's being empty and all 1's being all.
So, Ruzsa's theorem is about all, but would be, "larger" the set, then
that the researchers figured out a way to "attain" to it, then for something
like, "Falting's purity", whether the, "almost everywhere", as it were,
is not a hypocritical assignment.
https://arxiv.org/abs/2306.13403
Hmm, more about "doubling", here as about usually "doubling spaces"
and "doubling measures", about Ramsey theory, about the usual notion
of *-distribution or equi-distribution, of the 0/1 sequences, and that
the _language_ of the sequences and _space_, of the sequences, are
not the same thing, for Cantor space and Square Cantor space,
and, "Borel versus Combinatorics".
I.e., otherwise people could point at Ruzsa's theorem, and, in
a sense, point out a corresponding incompatibility, and show
counterexamples to these above theorems.
I.e., the rigorous formalist would sort of note that what really
needs be is a theory where Vitali sets make for Dirichlet and Poincare
their real analytical character, and that Banach-Tarski has quite different
arrivals of the geometric (the space) and algebraic (the words),
about the entire equi-interpretability, of otherwise these sorts results.

Doubling, ..., also about quasi-invariant measure theory and for physics,
quite about the continuous and discrete, and they meet in the middle,
doubling, ....
How many Fields' medalists do you see?

I'm not saying that it's any important, but, I've heard of me. I even looked into me a little bit, ....

What's funny is that something like this Green-Tao theorem, for Terence Tao the Fields medalist
and if you're interested in his opinion I suggest you look into his descriptive account of the
"hard" and "soft" in analysis,

https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/

the Green-Tao theorem that there are unbounded arithmetic
progressions, thus the not equi-distributed, besides that there is the equi-distributed like
.0101..., about the Borel side, "almost-everywhere" or as often written "a.e.", the one way
almost none, the other way almost all, just that there are any, at all.

https://blog.computationalcomplexity.org/2023/12/do-we-need-to-formalize.html

Here it's that there are formalizations both ways, because, foundations the mathematical
has a missing central edifice an Aristotelean continuum, [0,1] in order as by increment,
contiguous the beads-on-a-strong and continuous. (Which I wrote up.)

Then, it's interesting that Erdos and Szemeredi and for example Bergelson and Oprocha
for quasi-invariant measure theory, that Erdos in the era of Erdos and Szemeredi the
great number theorist, have lots going on in that part of the field that's about half
of the part of the field that's this stage of the space of Borel versus Combinatorics.

It's a lot like considering the continuous vis-a-vis the discrete,

with starting with a space of 0/1 sequences, that range in order from 0 to 1.

So, for a certain class of mathematicians in the field, it's important.
And, they're coming at these concepts from many directions.

[Ross Finlayson]

Post by Ross Finlayson

Post by Ross Finlayson
"But Ruzsa’s theorem required the subgroup to be enormous.
Marton’s insight was to posit that rather than being contained
in one giant subgroup, A could be contained in a polynomial
number of cosets of a subgroup that is no bigger than the original set A."
-- https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/
(The Quanta Magazine's are usually puff-pieces, but often about
interesting things.)
Here for example about the integers, here and generally natural integers,
there's a usual consideration about sequences of 0's and 1's, representing
subsets of the integers, with all 0's being empty and all 1's being all.
So, Ruzsa's theorem is about all, but would be, "larger" the set, then
that the researchers figured out a way to "attain" to it, then for something
like, "Falting's purity", whether the, "almost everywhere", as it were,
is not a hypocritical assignment.
https://arxiv.org/abs/2306.13403
Hmm, more about "doubling", here as about usually "doubling spaces"
and "doubling measures", about Ramsey theory, about the usual notion
of *-distribution or equi-distribution, of the 0/1 sequences, and that
the _language_ of the sequences and _space_, of the sequences, are
not the same thing, for Cantor space and Square Cantor space,
and, "Borel versus Combinatorics".
I.e., otherwise people could point at Ruzsa's theorem, and, in
a sense, point out a corresponding incompatibility, and show
counterexamples to these above theorems.
I.e., the rigorous formalist would sort of note that what really
needs be is a theory where Vitali sets make for Dirichlet and Poincare
their real analytical character, and that Banach-Tarski has quite different
arrivals of the geometric (the space) and algebraic (the words),
about the entire equi-interpretability, of otherwise these sorts results.

Doubling, ..., also about quasi-invariant measure theory and for physics,
quite about the continuous and discrete, and they meet in the middle,
doubling, ....
How many Fields' medalists do you see?

I'm not saying that it's any important, but, I've heard of me. I even looked into me a little bit, ....
What's funny is that something like this Green-Tao theorem, for Terence Tao the Fields medalist
and if you're interested in his opinion I suggest you look into his descriptive account of the
"hard" and "soft" in analysis,
https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/
the Green-Tao theorem that there are unbounded arithmetic
progressions, thus the not equi-distributed, besides that there is the equi-distributed like
.0101..., about the Borel side, "almost-everywhere" or as often written "a.e.", the one way
almost none, the other way almost all, just that there are any, at all.
https://blog.computationalcomplexity.org/2023/12/do-we-need-to-formalize.html
Here it's that there are formalizations both ways, because, foundations the mathematical
has a missing central edifice an Aristotelean continuum, [0,1] in order as by increment,
contiguous the beads-on-a-strong and continuous. (Which I wrote up.)
Then, it's interesting that Erdos and Szemeredi and for example Bergelson and Oprocha
for quasi-invariant measure theory, that Erdos in the era of Erdos and Szemeredi the
great number theorist, have lots going on in that part of the field that's about half
of the part of the field that's this stage of the space of Borel versus Combinatorics.
It's a lot like considering the continuous vis-a-vis the discrete,
http://youtu.be/mgreCqD2gqo
with starting with a space of 0/1 sequences, that range in order from 0 to 1.
So, for a certain class of mathematicians in the field, it's important.
And, they're coming at these concepts from many directions.

Wow, that sure is a lot of [Fields medalists] for one [theorem].


...

[Ross Finlayson]

Post by Ross Finlayson

Post by Ross Finlayson

Post by Ross Finlayson
"But Ruzsa’s theorem required the subgroup to be enormous.
Marton’s insight was to posit that rather than being contained
in one giant subgroup, A could be contained in a polynomial
number of cosets of a subgroup that is no bigger than the original set A."
-- https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/
(The Quanta Magazine's are usually puff-pieces, but often about
interesting things.)
Here for example about the integers, here and generally natural integers,
there's a usual consideration about sequences of 0's and 1's, representing
subsets of the integers, with all 0's being empty and all 1's being all.
So, Ruzsa's theorem is about all, but would be, "larger" the set, then
that the researchers figured out a way to "attain" to it, then for something
like, "Falting's purity", whether the, "almost everywhere", as it were,
is not a hypocritical assignment.
https://arxiv.org/abs/2306.13403
Hmm, more about "doubling", here as about usually "doubling spaces"
and "doubling measures", about Ramsey theory, about the usual notion
of *-distribution or equi-distribution, of the 0/1 sequences, and that
the _language_ of the sequences and _space_, of the sequences, are
not the same thing, for Cantor space and Square Cantor space,
and, "Borel versus Combinatorics".
I.e., otherwise people could point at Ruzsa's theorem, and, in
a sense, point out a corresponding incompatibility, and show
counterexamples to these above theorems.
I.e., the rigorous formalist would sort of note that what really
needs be is a theory where Vitali sets make for Dirichlet and Poincare
their real analytical character, and that Banach-Tarski has quite different
arrivals of the geometric (the space) and algebraic (the words),
about the entire equi-interpretability, of otherwise these sorts results.

Doubling, ..., also about quasi-invariant measure theory and for physics,
quite about the continuous and discrete, and they meet in the middle,
doubling, ....
How many Fields' medalists do you see?

I'm not saying that it's any important, but, I've heard of me. I even looked into me a little bit, ....
What's funny is that something like this Green-Tao theorem, for Terence Tao the Fields medalist
and if you're interested in his opinion I suggest you look into his descriptive account of the
"hard" and "soft" in analysis,
https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/
the Green-Tao theorem that there are unbounded arithmetic
progressions, thus the not equi-distributed, besides that there is the equi-distributed like
.0101..., about the Borel side, "almost-everywhere" or as often written "a.e.", the one way
almost none, the other way almost all, just that there are any, at all.
https://blog.computationalcomplexity.org/2023/12/do-we-need-to-formalize.html
Here it's that there are formalizations both ways, because, foundations the mathematical
has a missing central edifice an Aristotelean continuum, [0,1] in order as by increment,
contiguous the beads-on-a-strong and continuous. (Which I wrote up.)
Then, it's interesting that Erdos and Szemeredi and for example Bergelson and Oprocha
for quasi-invariant measure theory, that Erdos in the era of Erdos and Szemeredi the
great number theorist, have lots going on in that part of the field that's about half
of the part of the field that's this stage of the space of Borel versus Combinatorics.
It's a lot like considering the continuous vis-a-vis the discrete,
http://youtu.be/mgreCqD2gqo
with starting with a space of 0/1 sequences, that range in order from 0 to 1.
So, for a certain class of mathematicians in the field, it's important.
And, they're coming at these concepts from many directions.

Wow, that sure is a lot of [Fields medalists] for one [theorem].
...

Some people don't know that there are various models of integers, with various laws
of large numbers, which altogether are a more replete system reflecting continuity laws,
that mathematics has.

So, some statements in Ramsey theory, thusly so ignorant or naive, actually result their
undecideability, then that there are these sorts conjectures: _that are true both ways_,
weak and strong in each other, which it's possible to address, from knowing more and
better continuum mechanics from definitions of continuity in foundations, for a modern
21st'century foundations, that's more than coat-tailing and bad wall-papering.

In the world, ....


...