Algebraic numbers close to both 0 and 1

Zagier, D.

doi:10.1090/S0025-5718-1993-1197513-9

Algebraic numbers close to both $0$ and $1$
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Math. Comp. 61 (1993), 485-491 Request permission

Abstract:

A recent theorem of Zhang asserts that \[ H(\alpha ) + H(1 - \alpha ) \geq C\] for all algebraic numbers $\alpha \ne 0,1, (1 \pm \sqrt { - 3} )/2$, and some constant $C > 0$. An elementary proof of this, with a sharp value for the constant, is given (the optimal value of C is $\tfrac {1}{2}\log (\tfrac {1}{2}(1 + \sqrt 5 )) = 0,2406 \ldots$, attained for eight values of $\alpha$) and generalizations to other curves are discussed.

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