Thomas Bloom Explained

Thomas Bloom
Nationality:	British
Workplaces:	University of Cambridge University of Oxford University of Bristol University of Manchester
Alma Mater:	University of Oxford University of Bristol
Doctoral Advisor:	Trevor Wooley
Academic Advisors:	Timothy Gowers
Awards:	Royal Society University Research Fellowship

Thomas F. Bloom is a mathematician, who is a Royal Society University Research Fellow at the University of Manchester.^[1] He works in arithmetic combinatorics and analytic number theory.

Education and career

Thomas did his undergraduate degree in Mathematics and Philosophy at Merton College, Oxford. He then went on to do his PhD in mathematics at the University of Bristol under the supervision of Trevor Wooley. After finishing his PhD, he was a Heilbronn Research Fellow at the University of Bristol. In 2018, he became a postdoctoral research fellow at the University of Cambridge with Timothy Gowers. In 2021, he joined the University of Oxford as a Research Fellow.^[2] Then, in 2024, he moved to the University of Manchester, where he also took on a Research Fellow position.

Research

In July 2020, Bloom and Sisask^[3] proved that any set such that

\sum _{n} (1)/(n) diverges must contain arithmetic progressions of length 3. This is the first non-trivial case of a conjecture of Erdős postulating that any such set must in fact contain arbitrarily long arithmetic progressions.^{[4] [5]}

In November 2020, in joint work with James Maynard,^[6] he improved the best-known bound for square-difference-free sets, showing that a set

A \subset [N] with no square difference has size at most (N)/(( log N) ^{clog}) for some c> 0 .

In December 2021, he proved ^[7] that any set

A \subset N of positive upper density contains a finite S \subset A  such that \sum _{n} (1)/(n) =1 .^[8] This answered a question of Erdős and Graham.^[9]

Notes and References

1. Web site: Thomas Bloom - Mathematical Institute . 2024-09-14.
2. Web site: Thomas Bloom . 2022-07-28 . thomasbloom.org.
3. Bloom . Thomas F. . Sisask . Olof . 2021-09-01 . Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions . 2007.03528. math.NT .
4. Web site: Spalding . Katie . Math Problem 3,500 Years In The Making Finally Gets A Solution . IFLScience . 11 March 2022 . 28 July 2022 . en.
5. Web site: Klarreich . Erica . Landmark Math Proof Clears Hurdle in Top Erdős Conjecture . Quanta Magazine . 28 July 2022 . en . 3 August 2020.
6. Bloom . Thomas F. . Maynard . James . A new upper bound for sets with no square differences . 24 February 2021. math.NT . 2011.13266 .
7. Bloom . Thomas F. . 2021-12-07 . On a density conjecture about unit fractions . 2112.03726v2. math.NT.
8. Web site: Cepelewicz . Jordana . 2022-03-09 . Math's 'Oldest Problem Ever' Gets a New Answer . 2022-07-28 . Quanta Magazine . en.
9. Web site: Erdos . P. . Graham . R. . Paul Erdős . Ronald Graham. Old and new problems and results in combinatorial number theory . L'Enseignement Mathématique. Université de Genève. Semantic Scholar . 1980 . 23 April 2024.