Abstract:
For fixed $\varepsilon>0$, the following inequality holds:
$$
\Bigl|\frac uv-\beta\Bigr|>C\exp(-(\ln H)^{2+\varepsilon})
$$
for all numbers $\beta$ belonging to a field $K$ of finite degree over $Q$. The constant $C>0$ does not depend on beta. $H$ is the height of beta. $\wp(u)$ and $\wp(v)$ are algebraic numbers, and $u/v$ is a transcendental number. $\wp(z)$ is the Weierstrass function with complex multiplication and algebraic invariants. The proof is ineffective.
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