Simultaneous Approximation to a Basis of a Real Numberfield
Wolfgang M. Schmidt · American Journal of Mathematics · 1966
0 < aXq + pi C(q) (1?i?n). It is known [1,4] that N(h;a, * ,a.,,) is for almost all (al,* *,an) in the sense of Lebesgue asymptotically equal to * (h) for large h. Hence one would expect this to be true of many given n-tuples, say of (7r', P,e). One would be more hesitant to expect this of (/31<, * ,8,l/) if 1, I *,n is a basis of a real numberfield of degree n + 1, because of the exceptional status of such n-tuples in diophantine approximations. In fact, N(h;l, ; / , 83n) =0 for all h if gf(q) = cq-lln and 0 < c < c0(P1y . ,,3n). But the asymptotic formula does hold under a further mild assumption on t(q): THEOREM 1. Let J(q), 1(h) be as before, and assume that