* . *
  • About
  • Advertise
  • Privacy & Policy
  • Contact
Sunday, July 27, 2025
Earth-News
  • Home
  • Business
  • Entertainment
    Top 5 entertainment news: Sandeep Reddy Vanga regrets trimming Animal’s runtime by 7-8 minutes, Akshay Ku – Times of India

    Top 5 Entertainment Highlights: Sandeep Reddy Vanga Reveals Why He Trimmed Animal’s Runtime by 7-8 Minutes, Plus Akshay Ku Updates

    Cote de Pablo reveals how Michael Weatherly used his soap opera roots to put her at ease in “NCIS” love scene – yahoo.com

    Cote de Pablo Reveals How Michael Weatherly’s Soap Opera Background Made Their “NCIS” Love Scene Easier

    City of Pelham announces entertainment district plans for former Oak Mountain Amphitheatre site – WVTM

    Pelham Unveils Exciting New Entertainment District Plans for Former Oak Mountain Amphitheatre Site

    Black Box Players presents ‘The Three Musketeers’ – CBS 19 News

    Experience the Adventure: Black Box Players Bring ‘The Three Musketeers’ to Life!

    AP Entertainment SummaryBrief at 1:51 p.m. EDT – Channel 3000

    Entertainment Highlights: Key Updates You Can’t Miss

    ‘Devil Wears Prada 2’ casts Anne Hathaway’s love interest replacing Adrian Grenier’s Nate – Entertainment Weekly

    Devil Wears Prada 2′ Casts New Love Interest for Anne Hathaway, Replacing Adrian Grenier’s Nate

  • General
  • Health
  • News

    Cracking the Code: Why China’s Economic Challenges Aren’t Shaking Markets, Unlike America’s” – Bloomberg

    Trump’s Narrow Window to Spread the Truth About Harris

    Trump’s Narrow Window to Spread the Truth About Harris

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Trending Tags

    • Trump Inauguration
    • United Stated
    • White House
    • Market Stories
    • Election Results
  • Science
  • Sports
  • Technology
    AI’s race in the dark with China – Axios

    The High-Stakes AI Race: Innovation and Competition in the Shadows

    Eagle Unveils Revolutionary X-Ray Technology at Pack Expo

    Validea’s Top Information Technology Stocks Based On Peter Lynch – 7/25/2025 – Nasdaq

    Validea’s Top Information Technology Stocks Based On Peter Lynch – 7/25/2025 – Nasdaq

    WhoFi: New surveillance technology can track people by how they disrupt Wi-Fi signals – Tech Xplore

    WhoFi: New surveillance technology can track people by how they disrupt Wi-Fi signals – Tech Xplore

    Google Cloud Announced as a Key Technology Partner for Odoo Connect 2025 in San Francisco – GlobeNewswire

    Google Cloud Announced as a Key Technology Partner for Odoo Connect 2025 in San Francisco – GlobeNewswire

    Behind the Screens: The Impact of Technology on Real Estate – TRREB

    Behind the Screens: How Technology is Transforming the Future of Real Estate

    Trending Tags

    • Nintendo Switch
    • CES 2017
    • Playstation 4 Pro
    • Mark Zuckerberg
No Result
View All Result
  • Home
  • Business
  • Entertainment
    Top 5 entertainment news: Sandeep Reddy Vanga regrets trimming Animal’s runtime by 7-8 minutes, Akshay Ku – Times of India

    Top 5 Entertainment Highlights: Sandeep Reddy Vanga Reveals Why He Trimmed Animal’s Runtime by 7-8 Minutes, Plus Akshay Ku Updates

    Cote de Pablo reveals how Michael Weatherly used his soap opera roots to put her at ease in “NCIS” love scene – yahoo.com

    Cote de Pablo Reveals How Michael Weatherly’s Soap Opera Background Made Their “NCIS” Love Scene Easier

    City of Pelham announces entertainment district plans for former Oak Mountain Amphitheatre site – WVTM

    Pelham Unveils Exciting New Entertainment District Plans for Former Oak Mountain Amphitheatre Site

    Black Box Players presents ‘The Three Musketeers’ – CBS 19 News

    Experience the Adventure: Black Box Players Bring ‘The Three Musketeers’ to Life!

    AP Entertainment SummaryBrief at 1:51 p.m. EDT – Channel 3000

    Entertainment Highlights: Key Updates You Can’t Miss

    ‘Devil Wears Prada 2’ casts Anne Hathaway’s love interest replacing Adrian Grenier’s Nate – Entertainment Weekly

    Devil Wears Prada 2′ Casts New Love Interest for Anne Hathaway, Replacing Adrian Grenier’s Nate

  • General
  • Health
  • News

    Cracking the Code: Why China’s Economic Challenges Aren’t Shaking Markets, Unlike America’s” – Bloomberg

    Trump’s Narrow Window to Spread the Truth About Harris

    Trump’s Narrow Window to Spread the Truth About Harris

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    Israel-Gaza war live updates: Hamas leader Ismail Haniyeh assassinated in Iran, group says

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    PAP Boss to Niger Delta Youths, Stay Away from the Protest

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Court Restricts Protests In Lagos To Freedom, Peace Park

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Fans React to Jazz Jennings’ Inspiring Weight Loss Journey

    Trending Tags

    • Trump Inauguration
    • United Stated
    • White House
    • Market Stories
    • Election Results
  • Science
  • Sports
  • Technology
    AI’s race in the dark with China – Axios

    The High-Stakes AI Race: Innovation and Competition in the Shadows

    Eagle Unveils Revolutionary X-Ray Technology at Pack Expo

    Validea’s Top Information Technology Stocks Based On Peter Lynch – 7/25/2025 – Nasdaq

    Validea’s Top Information Technology Stocks Based On Peter Lynch – 7/25/2025 – Nasdaq

    WhoFi: New surveillance technology can track people by how they disrupt Wi-Fi signals – Tech Xplore

    WhoFi: New surveillance technology can track people by how they disrupt Wi-Fi signals – Tech Xplore

    Google Cloud Announced as a Key Technology Partner for Odoo Connect 2025 in San Francisco – GlobeNewswire

    Google Cloud Announced as a Key Technology Partner for Odoo Connect 2025 in San Francisco – GlobeNewswire

    Behind the Screens: The Impact of Technology on Real Estate – TRREB

    Behind the Screens: How Technology is Transforming the Future of Real Estate

    Trending Tags

    • Nintendo Switch
    • CES 2017
    • Playstation 4 Pro
    • Mark Zuckerberg
No Result
View All Result
Earth-News
No Result
View All Result
Home Science

‘A-Team’ of Math Proves a Critical Link Between Addition and Sets

December 6, 2023
in Science
‘A-Team’ of Math Proves a Critical Link Between Addition and Sets
Share on FacebookShare on Twitter

The lists have fixed lengths, and each bit can be either 0 or 1. You add them together by adding each entry to its counterpart in another list, with the rule that 1 + 1=0. So (0, 1, 1, 1, 0) + (1, 1, 1, 1, 1)=(1, 0, 0, 0, 1). PFR is an attempt to figure out what a set can look like if it isn’t quite a subgroup but has some grouplike features.

To make PFR precise, imagine you have a set of binary lists called A. Now take every pair of elements from A and add them up. The resulting sums make up the sumset of A, called A + A. If the elements of A are chosen randomly, then most of the sums are different from each other. If there are k elements in A, that means there will be around k2/2 elements in the sumset. When k is large — say, 1,000 — k2/2 is a lot bigger than k. But if A is a subgroup, every element of A + A is in A, meaning that A + A is the same size as A itself.

PFR considers sets that are not random, but are not subgroups either. In these sets, the number of elements in A + A is somewhat small — say, 10k, or 100k. “It’s really useful when your notion of structure is much more rich than just being an exact algebraic structure,” said Shachar Lovett, a computer scientist at the University of California, San Diego.

All the sets mathematicians knew of that obeyed this property “are pretty close to actual subgroups,” Tao said. “That was the intuition, that there weren’t any other sort of fake groups lying around.” Freiman had proved a version of this statement in his original work. In 1999, Ruzsa extended Freiman’s theorem from the integers to the setting of binary lists. He proved that when the number of elements in A + A is a constant multiple of the size of A, A is contained within a subgroup.

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.

‘I Know a Real Idea When I See a Real Idea’

Around the turn of the millennium, Gowers came across Ruzsa’s proofs of Freiman’s theorem while studying a different problem about sets containing strings of evenly spaced numbers. “I needed something like this, kind of getting structural information from much looser information about a certain set,” Gowers said. “I was very fortunate that not that long before, Ruzsa produced this absolutely gorgeous proof.”

Gowers began to work out a potential proof of the polynomial version of the conjecture. His idea was to start with a set A whose sumset was relatively small, then gradually manipulate A into a subgroup. If he could prove that the resulting subgroup was similar to the original set A, he could easily conclude the conjecture was true. Gowers shared his ideas with colleagues, but no one could mold them into a full proof. Though Gowers’ strategy was successful in some cases, in others the manipulations took A further away from the desired conclusion of the polynomial Freiman-Ruzsa conjecture.

Eventually, the field moved on. In 2012, Sanders almost proved PFR. But the number of shifted subgroups he needed was above the polynomial level, though only by a little bit. “Once he did that, it meant that it became a less urgent thing, but still a really nice problem for which I have a great fondness,” Gowers said.

But Gowers’ ideas lived on in his colleagues’ memories and hard drives. “There’s a real idea there,” said Green, who had also been a student of Gowers. “I know a real idea when I see a real idea.” This summer, Green, Manners and Tao finally liberated Gowers’ ideas from their purgatory.

Green, Tao and Manners were 37 pages deep in collaboration before they thought to return to Gowers’ 20-year-old ideas. In a June 23 paper, they had successfully used a concept from probability theory called random variables to probe the structure of sets with small sumsets. By making this switch, the group could manipulate their sets with more finesse. “Dealing with random variables is somehow much less rigid than dealing with sets,” Manners said. With a random variable, “I can tweak one of the probabilities by a small amount, and that might give me a better random variable.”

Using this probabilistic perspective, Green, Manners and Tao could move from working with the number of elements in a set to a measurement of the information contained in a random variable, a quantity called entropy. Entropy was not new to additive combinatorics. In fact, Tao had attempted to popularize the concept in the late 2000s. But nobody had yet tried to use it on the polynomial Freiman-Ruzsa conjecture. Green, Manners and Tao discovered it was powerful. But they still couldn’t prove the conjecture.

As the group chewed over their new results, they realized that they’d finally built an environment in which Gowers’ dormant ideas could flourish. If they measured the size of a set using its entropy, rather than its number of elements, the technical details might work out much better. “At some point we realized that these old ideas from Tim from 20 years ago were actually more likely to work than the ones we were trying,” Tao said. “And so we brought Tim back into the project. And then all the pieces fit together surprisingly nicely.”

Still, there were many details to figure out before a proof came together. “It was sort of foolish that all four of us were incredibly busy with other things,” Manners said. “You want to publish this great result and tell the world, but you do actually still have to write your midterms.” Eventually, the group pushed through, and on November 9, they posted their paper. They proved that if A + A is no bigger than k times the size of A, then A can be covered by no more than about k12 shifts of a subgroup that is no bigger than A. This is a potentially enormous number of shifts. But it is a polynomial, so it doesn’t grow exponentially faster as k gets bigger, as it would if k were in the exponent.

A few days later, Tao began to formalize the proof. He ran the formalization project collaboratively, using the version-control package GitHub to coordinate contributions from 25 volunteers around the world. They used a tool called Blueprint developed by Patrick Massot, a mathematician at Paris-Saclay University, to organize the efforts to translate from what Tao called “Mathematical English” into computer code. Blueprint can, among other things, create a chart depicting the various logical steps involved in the proof. Once all the bubbles were covered in what Tao called a “lovely shade of green,” the team was done. They discovered a few very minor typos in the paper — in an online message, Tao noted that “the most mathematically interesting portions of the project were relatively straightforward to formalize, but it was the technical ‘obvious’ steps that took the longest.”

Marton died just a few years before her famous conjecture was proved, but the proof echoes her life’s work on entropy and information theory. “Everything works much better when you do it in this entropy framework than in the framework I was trying to do it,” Gowers said. “To me, it still seems somewhat magical.”

Quanta is conducting a series of surveys to better serve our audience. Take our mathematics reader survey and you will be entered to win free Quanta merch.

>>> Read full article>>>
Copyright for syndicated content belongs to the linked Source : Quanta Magazine – https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/

Tags: ‘A-Team’Provesscience
Previous Post

WHO calls on countries to increase taxes on alcohol and sugary sweetened beverages

Next Post

Revolutionizing Nanotechnology: Photonic Cavities that Self-Assemble at the Atomic Level

AI’s race in the dark with China – Axios

The High-Stakes AI Race: Innovation and Competition in the Shadows

July 27, 2025
LeBron James’ future with Lakers beyond 2025-26 certain to generate speculation leading up to training camp – Yahoo Sports

LeBron James’ Lakers Future Beyond 2025-26 Sparks Intense Speculation Ahead of Training Camp

July 27, 2025
Energy landscapes direct the movement preferences of elephants – besjournals

Energy landscapes direct the movement preferences of elephants – besjournals

July 27, 2025
At United Nations Summer Program, Computer Science Student Examines AI Ethics – Georgetown University

At United Nations Summer Program, Computer Science Student Examines AI Ethics – Georgetown University

July 27, 2025
Return of wolves to Yellowstone has led to a surge in aspen trees unseen for 80 years – Live Science

Wolves’ Return to Yellowstone Sparks Unprecedented Aspen Tree Boom After 80 Years

July 27, 2025
‘Bluey’ Is Growing Up: BBC Plots All-Ages Expansion For The Franchise As A Full-Fledged Lifestyle Brand – Cartoon Brew

‘Bluey’ Is Growing Up: BBC Plots All-Ages Expansion For The Franchise As A Full-Fledged Lifestyle Brand – Cartoon Brew

July 27, 2025
Katie Ledecky, Summer McIntosh and a race for the ages at World Swimming Championships – NBC Sports

Katie Ledecky and Summer McIntosh Ignite an Epic Showdown at the World Swimming Championships

July 27, 2025
5 ways Trump has shaped the economy in 6 months – The Hill

5 Game-Changing Ways Trump Transformed the Economy in Just 6 Months

July 27, 2025
Flutter Entertainment Announces Pricing Of $1.272 Bln Of Additional Senior Secured Notes Due 2031 – Nasdaq

Flutter Entertainment Raises $1.27 Billion in Major Senior Secured Notes Offering Due 2031

July 27, 2025
KFF Health Tracking Poll: Public Views on Recent Tax and Budget Legislation – KFF

How the Public Really Feels About the Latest Tax and Budget Changes

July 27, 2025

Categories

Archives

July 2025
MTWTFSS
 123456
78910111213
14151617181920
21222324252627
28293031 
« Jun    
Earth-News.info

The Earth News is an independent English-language daily published Website from all around the World News

Browse by Category

  • Business (20,132)
  • Ecology (741)
  • Economy (765)
  • Entertainment (21,645)
  • General (16,135)
  • Health (9,803)
  • Lifestyle (773)
  • News (22,149)
  • People (767)
  • Politics (774)
  • Science (15,980)
  • Sports (21,263)
  • Technology (15,747)
  • World (748)

Recent News

AI’s race in the dark with China – Axios

The High-Stakes AI Race: Innovation and Competition in the Shadows

July 27, 2025
LeBron James’ future with Lakers beyond 2025-26 certain to generate speculation leading up to training camp – Yahoo Sports

LeBron James’ Lakers Future Beyond 2025-26 Sparks Intense Speculation Ahead of Training Camp

July 27, 2025
  • About
  • Advertise
  • Privacy & Policy
  • Contact

© 2023 earth-news.info

No Result
View All Result

© 2023 earth-news.info

No Result
View All Result

© 2023 earth-news.info

Go to mobile version