Thm (Kronecker).- If all conjugates of an algebraic integer lie on the unit circle, then the integer is a root of unity.
Question: Can one provide a good effective version of this? That is: given that we have an algebraic integer alpha of degree <=d, can we show that alpha has a conjugate that is at least epsilon away from the unit circle, where epsilon depends only on d? It actually isn't hard to do this (from the standard proof of Kronecker, viz.: alpha, alpha^2, alpha^3… are all algebraic integers, and their minimal polynomials would eventually repeat (being bounded) if all conjugates of alpha lied on the unit circle) with
epsilon exponential on d, i.e., epsilon of the form epsilon = 1/C^d; what we actually want is an epsilon of the form 1/d^C, say.
(Question really due to B. Bukh.)
Best Answer
If $M(\alpha)$ is the Mahler measure of $\alpha$, then the largest conjugate of $\alpha$ has absolute value at least
$$1 + \frac{\log(M(\alpha))}{d}.$$
Lehmer's conjecture implies that this at least $O(d^{-1})$ away from one. Dobrowolski's lower bound for $M(\alpha)$ shows that there is a conjugate at least $O(d^{-1-\epsilon})$ away from $1$ for any $\epsilon > 0$. Better bounds are available if one has more information, for example, the signature of $\mathbf{Q}(\alpha)$, or whether $\alpha$ is conjugate to $\alpha^{-1}$ or not.
For explicit references, see
http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf