Let $x$ be an algebraic number. Must $\arctan(x)/\pi$ have finite irrationality measure? Are there any useful upper bounds?



Upper bounds on the irrationality measure of the arctan of an algebraic number
Let $\alpha=\frac{1+xi}{\sqrt{1+x^2}}$. There are some cases $(\arctan x) /\pi$ is rational. For example, $x=1, \sqrt3$. In these cases, $(\arctan x)/\pi$ has the irrationality measure $1$. These occur precisely when $\alpha$ is a root of unity.


Since $x$ is algebraic, so is $\alpha$. Then $\arctan x = \arg \alpha = (\log \alpha)/i$. Using $\log(-1)=i\pi$, we have $(\arctan x)/\pi= (\log \alpha)/\log (-1)$. Here, we need a fixed determination of logarithms of complex numbers.


Suppose that $\alpha$ is not a root of unity. We must have that $(\arctan x)/\pi$ is irrational. That is, there is no nonzero rational solutions to $\beta\_1 \log \alpha + \beta\_2 \log(-1) = 0$. By Gelfond-Schneider theorem, $(\log \alpha)/\log(-1)$ is transcendental. Thus, the irrationality measure of $(\arctan x)/\pi$ is at least $2$. Note that the transcendence of $(\arctan x)/\pi$ also follows from the argument below.


For the upper bound of the irrationality measure, we apply [Baker-Wustholz theorem](https://assets.cambridge.org/97805218/07999/excerpt/9780521807999_excerpt.pdf). The logarithmic form $$L=\beta\_1 \log\alpha + \beta\_2 \log (-1)$$ is nonvanishing for any pair of integers $(\beta\_1,\beta\_2)\neq (0,0)$. Let $n=2$ and let $d$ be the degree of $\alpha$ over $\mathbb{Q}$. Then the theorem yields
$$
\log |L| > -C(2,d)h'(\alpha)h'(-1)h'(L)
$$
where
$C(2,d)= 905969664 \ d^4\log(4d)$, $h'(-1)=1$, $h'(\alpha)$ is a modified height of $\alpha$ defined by
$$
h'(\alpha)=\frac1d \max(h(\alpha), |\log \alpha|, 1)
$$
and $h(\alpha)$ is the logarithmic Weil height of $\alpha$. We may use $h'(L)=\log (\max(|\beta\_1|/b, |\beta\_2|/b))+1$ although the definition is slightly different in the note. Also, $b=\gcd (\beta\_1,\beta\_2)$.


Hence, there are positive numbers $A(d,\alpha)$, $B(d,\alpha)$ such that for any integer $p$ and any positive integer $q$,
$$
\left|\frac{\log \alpha}{\log(-1)}- \frac pq \right|> \frac{A(d,\alpha)}{q^{B(d,\alpha)}}.
$$


As we see above, the number $B(d,\alpha)$ is quite huge, but it gives an upper bound of the irrationality measure of $(\arctan x)/\pi$.


