Sequences of Diophantine approximations

AbstractWe sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of “good” Diophantine approximations to a fixed α ∈ C are trivial ones. For example, suppose that a > 1 and that (δn)n=1∞ and (σn)n=1∞ are two positive, strictly increasing unbounded sequences satisfying δn+1 ≤ aδn and σn+1 ≤ aσn. If there is a sequence of nonzero polynomials Pn ∈ Z[x] with deg Pn ≤ δn, deg Pn + log height Pn ≤ σn, and ∣Pn(α)∣ ≤ e−(2a+1)δnσn, then each Pn(α) = 0