@MISC{DIOPHAN2.MIZ,
  AUTHOR = {Watase, Yasushige},
  TITLE = {Introduction to {D}iophantine Approximation -- Part 2},
  SECTION1 = {Preliminaries},
  SECTION2 = {Hurwitz' theorem (1891)},
  SECTION3 = {Minkowski's theorem (1904)},
  DAY = {29}, 
  MONTH = {November},
  YEAR = {2017},
  ADDRESS1 = {Suginami-ku Matsunoki 3 Tokyo\\Japan},
  SUMMARY = {This article provides formalized proofs for Hurwitz' theorem (1891) and
  Minkowski's theorem (1904). Both theorems are well known in
  the theory of Diophantine approximations
  as appeared in \cite{HardyWright:2008}\cite{NIVEN:2008}.
  \indent  A formal proof of Dirichlet's theorem, namely an inequation $| \theta - y/x | \leq  1/x^2$
  has infinitely many integer solutions $(x,y)$ where $\theta$ is an irrational number, was given in FM23(2).
  A finer approximation is given by Hurwitz' theorem: $| \theta - y/x | \leq  1/\sqrt5 x^2$ .\\
  \indent Minkowski's theorem concerns about an inequation of
  a product of non-homogeneous binary linear forms such that
  $| a_1 x + b_1 y + c_1 |\cdot | a_1 x + b_1 y + c_1 | \leq \Delta/4$
  where $\Delta = |a_1 b_2 - a_2 b_1| \neq 0$, has at least
  one integer solution.}}

@BOOK{MINKOWSKI:1907,
 AUTHOR={H. Minkowski},
 TITLE={Diophantische Approximationen. Einfuhrung in die Zahlentheorie},
 PUBLISHER={Taubner, Leipzig},
 YEAR=1907}

@BOOK{HardyWright:2008,
 AUTHOR={G. H. Hardy and E. M. Wright},
 TITLE={An Introduction to The Theory of Numbers 6th edition},
 PUBLISHER={Oxford University Press},
 YEAR=2008}

@BOOK{NIVEN:2008,
 AUTHOR={Ivan, Niven},
 TITLE={Diophantine Approximation},
 PUBLISHER={Dover},
 YEAR=2008}

