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


      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' In 1963 as a graduate student, he showed that if the positive integers are finitely colored, then some color class is piece-wise syndetic.
      In A Density Version of a Geometric Ramsey Theorem, he and Joe P. Buhler showed that “for every



      ε
      >
      0


      {\displaystyle \varepsilon >0}

      there is an



      n
      (
      ε
      )


      {\displaystyle n(\varepsilon )}

      such that if



      n
      =
      d
      i
      m
      (
      V
      )
      ≥
      n
      (
      ε
      )


      {\displaystyle n=dim(V)\geq n(\varepsilon )}

      then any subset of



      V


      {\displaystyle V}

      with more than



      ε

      |

      V

      |



      {\displaystyle \varepsilon |V|}

      elements must contain 3 collinear points” where



      V


      {\displaystyle V}

      is an



      n


      {\displaystyle n}

      -dimensional affine space over the field with




      p

      d




      {\displaystyle p^{d}}

      elements, and



      p
      ≠
      2


      {\displaystyle p\neq 2}

      ".
      In Descriptions of the characteristic sequence of an irrational, Brown discusses the following idea: Let



      α


      {\displaystyle \alpha }

      be a positive irrational real number. The characteristic sequence of



      α


      {\displaystyle \alpha }

      is



      f
      (
      α
      )
      =

      f

      1



      f

      2


      …


      {\displaystyle f(\alpha )=f_{1}f_{2}\ldots }

      ; where




      f

      n


      =
      [
      (
      n
      +
      1
      )
      α
      ]
      [
      α
      ]


      {\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



      [
      n
      α
      ]


      {\displaystyle [n\alpha ]}

      .” He then gives some conclusions regarding the conditions for



      [
      n
      α
      ]


      {\displaystyle [n\alpha ]}

      which are equivalent to




      f

      n


      =
      1


      {\displaystyle f_{n}=1}

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


      References




      External links


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

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