对Baker~[1]-Mahler~[2]一个定理的注记

A NOTE ON A THEOREM OF BAKER-MAHLER

对任意实数x,定义‖x‖=max(x-[x],[x]+1-x).设a_1,…,a_(k-1)是互不相等的非零整数,a是适合(a,a_1,…,a_(k-1)=1的正整数,r是正整数.

Let k ≥ 3 and a_1,…, a_(k-1) be distinct non-vanishing integers. Let a be a positive integer such that (a, a_1,…, a_(k-1)) = 1. Further, let B = a + max(|a_1|, …,|a_(k-1)|) and E_i = e^(ai/a)( 1≤i≤k - 1).We haveTheorem. Suppose that y ≥B^(16k4)·B^(16k4) Then we have y‖yE_1‖…‖yE_(k-1)‖>y^(-12k2(k+1)(log B·(log log y)-1))1/2(1)This gives a slight modification of a theorem due to Mahler, whose original result was obtained by replacing the right-hand side of (1) with y^(-12k3(k-1) (log B·(log log y)-1)1/2