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The well-known Liouville theorem asserts that an irrational algebraic number $\alpha$ cannot have too good rational approximations, namely $|\alpha-p/q|\ge C(\alpha)/q^k$ where $k$ is the degree of $\alpha$. I wonder whether a similar result holds for the argument of an algebraic number that happens to lie on the unit circle.

For example, consider $\theta=\arccos(1/3)$. It is the argument of a root $\alpha$ of the polynomial $x^2-\frac23x+1$. Is there a similar lower bound for $|\theta/\pi-p/q|$, or equivalently, for $|\alpha^q-1|$? (Note that $|\alpha^q-1|\approx q\cdot |\theta/2\pi-p/q|$).

More generally, let $\alpha\in\mathbb C$ be an algebraic number, $|\alpha|=1$ and $\alpha$ is not a root of unity. Is it always true that $|\alpha^q-1|\ge C(\alpha)/q^k$ where $k$ depends only on the degree of $\alpha$ (or perhaps equals the degree minus one)?

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  • $\begingroup$ Sergei, Shouldn't both of your $\le$ signs be $\ge$'s? $\endgroup$
    – Sidney Raffer
    Commented Apr 19, 2011 at 10:57
  • $\begingroup$ @Sergei: I asked Győry who asked Waldschmidt. They think your question is hard, and they don't know the answer. So it is very likely that your question is an open problem. $\endgroup$
    – GH from MO
    Commented Apr 26, 2011 at 22:38

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The answer is affirmative by a result of Fel'dman: An improvement of the estimate of a linear form in the logarithms of algebraic numbers (Russian), Mat. Sb. (N.S.) 77 (119) 1968, 423–436, MR0232736.

See also Theorem 3.1 in Baker's Transcendental Number Theory.

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    $\begingroup$ Can you elaborate on how to apply Feldman's result here? I could only get something like $C(\alpha)/q^{\log q}$ from the first theorem cited on the MR page, and an estimate exponential in $q$ from the second one. $\endgroup$
    – Sergei Ivanov
    Commented Apr 19, 2011 at 16:42
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    $\begingroup$ @Sergei: In the second theorem cited on the MR page, $m=2$, $H_0=q$, while $h$ and $n$ are positive constants depending on $\alpha$, so the bound is of the form $>\exp(-c_1-c_2*\ln q)=c_3 q^{-c_2}$ with constants $c_i>0$ depending on $\alpha$. $\endgroup$
    – GH from MO
    Commented Apr 19, 2011 at 17:11
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    $\begingroup$ Thanks, I think I got it now. Do you know if the exponent can be bounded in terms of degree of $\alpha$ (not involving height or whatever)? $\endgroup$
    – Sergei Ivanov
    Commented Apr 19, 2011 at 18:32
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    $\begingroup$ @Sergei: Unfortunately I don't know, so I was not completely right saying "the answer is affirmative". The state of the art seems to be set by Matveev (MR1817252) improving on the previous strong result by Baker-Wüstholz (MR1234835). The latter is available online: digizeitschriften.de/dms/img/… I would probably ask the authors of these papers how significant is the dependence on the heights. Probably this dependence is substantial even for $\alpha$ on the unit circle, but I am no expert. $\endgroup$
    – GH from MO
    Commented Apr 19, 2011 at 19:40

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