Upper bounds on the irrationality measure of the arctan of an algebraic number

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

• What if $x=\sqrt{2}$? Mar 10, 2021 at 2:22
• I don't know. Is there a reason $\sqrt{2}$ is special here? Mar 10, 2021 at 4:29
• Yes. The number has finite irrationality measure. A bound is obtained from Baker-Wustholtz theorem, but probably very big. en.wikipedia.org/wiki/Baker%27s_theorem Mar 10, 2021 at 5:08
• Yes they are. The bound will depend on the degree of the algebraic number x. Mar 10, 2021 at 5:40
• @SungjinKim: perhaps you can convert your comment into an answer? Could you explain how does it follow from Baker's Theorem? Mar 10, 2021 at 6:48

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. 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 $$(\arctan x)/\pi$$.