Approximation to real numbers by cubic algebraic integers. II


It has been conjectured for some time that, for any integer $n\ge 2$, any real number $\varepsilon >0$ and any transcendental real number $\xi$, there would exist infinitely many algebraic integers $\alpha$ of degree at most $n$ with the property that $|\xi-\alpha|\le H(\alpha)^{-n+\varepsilon}$, where $H(\alpha)$ denotes the height of $\alpha$. Although this is true for $n=2$, we show here that, for $n=3$, the optimal exponent of approximation is not $3$ but $(3+\sqrt{5})/2\simeq 2.618$.


Damien Roy

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5