Dirichlet's diophantine approximation theorem

T.W. Cusick
1977 Bulletin of the Australian Mathematical Society  
One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if a" a_, ..., a are any real numbers and m 2 2 is an integer, then there exist integers q, p , p , ..., p such that 1 5 q < m and |qa.-p. | £ mh olds for 1 £ i £ n . The paper considers the problem of the extent to which this theorem can be improved by replacing mb y a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler {.Bull. Austral. Math. Soc
more » ... Austral. Math. Soc 14 (1976), 1+63-U65] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved. For any real number x , let ||x|| denote the distance from x to the nearest integer; thus % > ||x|| > 0 for all x . The well-known theorem of Dirichlet concerning simultaneous diophantine approximation can be stated as follows: if a. , a ? , ..., a are any real numbers and m > 2 is an integer, then there exists an integer q such that (1) 1 £ q < m and ||qct.|| < m~1 /n
doi:10.1017/s0004972700023224 fatcat:g2j5pd4ckrf57lj4vrj55xtnru