Algebraic numbers close to both 0 and 1

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