Tom Brown (mathematician)

American-Canadian mathematician

Thomas Craig Brown (born 1938) is an American-Canadian mathematician, Ramsey Theorist, and Professor Emeritus at Simon Fraser University.^[1]

Collaborations

As a mathematician, Brown’s primary focus in his research is in the field of Ramsey Theory. When completing his Ph.D., his thesis was 'On Semigroups which are Unions of Periodic Groups'^[2] In 1963 as a graduate student, he showed that if the positive integers are finitely colored, then some color class is piece-wise syndetic.^[3]

In A Density Version of a Geometric Ramsey Theorem.^[4] he and Joe P. Buhler show that “for every ${\displaystyle \varepsilon >0}$ there is an ${\displaystyle n(\varepsilon )}$ such that if ${\displaystyle n=dim(V)\geq n(\varepsilon )}$ then any subset of ${\displaystyle V}$ with more than ${\displaystyle \varepsilon |V|}$ elements must contain 3 collinear points” where ${\displaystyle V}$ is an ${\displaystyle n}$-dimensional affine space over the field with ${\displaystyle p^{d}}$ elements, and ${\displaystyle p\neq 2}$".

In Descriptions of the characteristic sequence of an irrational,^[5] Brown discusses the following idea: Let ${\displaystyle \alpha }$ be a positive irrational real number. The characteristic sequence of ${\displaystyle \alpha }$ is ${\displaystyle f(\alpha )=f_{1}f_{2}\ldots }$; where ${\displaystyle f_{n}=[(n+1)\alpha ][\alpha ]}$.” From here he discusses “the various descriptions of the characteristic sequence of α which have appeared in the literature” and refines this description to “obtain a very simple derivation of an arithmetic expression for ${\displaystyle [n\alpha ]}$.” He then gives some conclusions regarding the conditions for ${\displaystyle [n\alpha ]}$ which are equivalent to ${\displaystyle f_{n}=1}$.

He has collaborated with Paul Erdős, including Quasi-Progressions and Descending Waves^[6] and Quantitative Forms of a Theorem of Hilbert.^[7]

References

1. "Tom Brown Professor Emeritus at SFU". Retrieved 10 November 2020.
2. Jensen, Gary R.; Krantz, Steven G. (2006). 150 Years of Mathematics at Washington University in St. Louis. American Mathematical Society. p. 15. ISBN 978-0-8218-3603-3.
3. Brown, T. C. (1971). "An interesting combinatorial method in the theory of locally finite semigroups" (PDF). Pacific Journal of Mathematics. 36 (2): 285–289. doi:10.2140/pjm.1971.36.285.
4. Brown, T. C.; Buhler, J. P. (1982). "A Density version of a Geometric Ramsey Theorem" (PDF). Journal of Combinatorial Theory. Series A. 32: 20–34. doi:10.1016/0097-3165(82)90062-0.
5. Brown, T. C. (1993). "Descriptions of the Characteristic Sequence of an Irrational" (PDF). Canadian Mathematical Bulletin. 36: 15–21. doi:10.4153/CMB-1993-003-6.
6. Brown, T. C.; Erdős, P.; Freedman, A. R. (1990). "Quasi-Progressions and Descending Waves". Journal of Combinatorial Theory. Series A. 53: 81–95. doi:10.1016/0097-3165(90)90021-N.
7. Brown, T. C.; Chung, F. R. K.; Erdős, P. (1985). "Quantitative Forms of a Theorem of Hilbert" (PDF). Journal of Combinatorial Theory. Series A. 38 (2): 210–216. doi:10.1016/0097-3165(85)90071-8.

External links

• Archive of papers published by Tom Brown
• 2003 INTEGERS conference dedicated to Tom Brown