engleski

On p-adic T-numbers

For a transcendental number x in Qp, denote by wn(x) the upper limit of the real numbers w for which there exist infinitely many integer polynomials P(X) of degree at most n satisfying 0 < |P(x)|_p <= H(P)^{; ; ; -w-1}; ; ; . Also, denote by w*n(x) the upper limit of the real numbers w for which there exist infinitely many algebraic numbers alpha in Qp of degree at most n satisfying 0 < |x-alpha|_p <=  H(alpha)^{; ; ; -w-1}; ; ; . We prove the following result: Let (wn) and (w*n) be two non-decreasing sequences in [1, infty] such that w*n <= wm <= w*n + (n-1)/n, wn>n^3+2n^2+5n+2, for any n>=1. Then there exists a p-adic transcendental number x such that w*n(x)=w*n and wn(x)=wn, for any n>=1.

p-adic numbers ; T-numbers.

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