Simultaneous approximation to algebraic numbers by rationals
Wolfgang M. Schmidt · Acta Mathematica · 1970
We shall prove theorems on simultaneous approximation which generalize Roth's well-known theorem [3] on rational approximation to a single algebraic irrational ~.Throughout the paper, []~]1 will denote the distance from a real number ~ to the nearest integer.TH~.OREM 1.Let ax ..... an be real algebraic numbers such that 1, al ..... 0~ are linearly independent over the field Q o/rationals.Then/or every e > 0 there are only finitely many positive integers q with COROLLARY.Suppose o~ 1 .... , o:~, ~ are as above.There are only finitely many n-tuples (Pl/q ..... p~/q) o/rationals satis/yin9 I~ ~ _ (pjq) 2 ..... n).(2)A dual to Theorem 1 is as follows.THEORE~ 2. Let al ..... an, e be as in Theorem 1.There are only finitely many n-tuplss o/nonzero integers ql .... , qn with Ilq,~, +... + q,~.ll"Iq, q~... q,t 1+" o and with Iq~+ ...