Tom Brown (mathematician)

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]

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