Convergents of continued fractions provide in some sense the best rational approximations to an irrational number. In my research trying to prove that the binary digits of $\sqrt{2}$ are 50% one, 50% zero, I came up with something interesting.
I analyzed the successive records (minima) of $w_n = |\{n\alpha\} - \beta|$ where $\{n\alpha\}$ is $n\alpha$ modulo $1$ (the brackets represent the fractional part function,) $n =1, 2$ and so on, $\beta = 1/2$ and $\alpha = \log_2 3$. I found how the arrival times $t_n$ of these records are related to the denominators of the convergents of $\alpha$. More specifically, the $\Delta t_n$ are linked to these convergents, and also relevant to musical theory. My findings are summarized here.
I did some more reseach and found that something rather similar happens with $\beta = e^{-1}$, and probably with almost all $\beta \in [0, 1]$. I posted it as an exercise:
Exercise:
Let $\alpha = \log_2 3$ and $w_n = | \{ n\alpha \} - 1/2 |$. Find the arrival times of the first records for $w_n$, focusing on minima rather than maxima. Let $t_n$ denotes the arrival time of the $n$-th record. The sequence $\Delta t_n = t_n - t_{n-1}$ is a subsequence of the one listed here.
Also,
- $\Delta t_6 = \Delta t_7 = \cdots = \Delta t_{17} = 665$, and
- $\Delta t_{20} = \Delta t_{21} = \cdots = \Delta t_{47} = 190537$.
Now do the same with $w_n = | \{n \alpha\} - 1/e |$. The same $\Delta t_n$ show up, in particular
- $\Delta t_9 = \Delta t_{10} = \cdots = \Delta t_{17} = 665$, and
- $\Delta t_{19} = \Delta t_{20} = \cdots = \Delta t_{38} = 190537$.
However $\Delta t_8 = 4296$ is unusually large. Is there a pattern here? Does it generalize to any $\beta \in [0, 1]$? Is there an explanation for this?
Note
This problem can be re-phrased as follows: we are trying to find the best approximation of the form $p\alpha - q$ for the number $\beta$, with $p, q$ being strictly positive integers. The $n$-th best approximant for $\beta$ is $\{t_n \alpha\}$, and of course $\{t_n\alpha\} \rightarrow \beta$.