The Universe of Discourse

Thu, 22 Jul 2021

The convergents of 2x

Take some real number !!\alpha!! and let its convergents be !!c_0, c_1, c_2, \ldots!!. Now consider the convergents of !!2\alpha!!. Sometimes they will include !!2c_0, 2c_1, 2c_2, \ldots!!, sometimes only some of these.

For example, the convergents of !!\pi!! and !!2\pi!! are

$$ \begin{array}{rlc} \pi & \approx & \color{darkblue}{3},&&& \color{darkblue}{\frac{22}{7}}, & \color{darkblue}{\frac{333}{106}}, && \color{darkblue}{\frac{355}{113}}, & \color{darkblue}{\frac{103993}{33102}}, && \frac{104348}{33215}, & \color{darkblue}{\frac{208341}{66317}}, & \ldots \\ 2\pi & \approx & \color{darkblue}{6}, & \frac{19}{3}, & \frac{25}{4}, & \color{darkblue}{\frac{44}{7}}, & \color{darkblue}{\frac{333}{53}}, & \frac{377}{60}, & \color{darkblue}{\frac{710}{113}}, & \color{darkblue}{\frac{103393}{16551}}, & \frac{312689}{49766}, && \color{darkblue}{\frac{416682}{66317}}, & \ldots \end{array}

Here are the analogous lists for !!\frac{1+\sqrt{5}}2!! and !!1+\sqrt5!!:

$$ \begin{array}{rlc} \frac12{1+\sqrt{5}}& \approx & 1, & 2, & \color{darkblue}{\frac32}, & \frac53, & \frac85, & \color{darkblue}{\frac{13}8}, & \frac{21}{13}, & \frac{34}{21}, & \color{darkblue}{\frac{55}{34}}, & \frac{89}{55}, & \frac{144}{89}, & \color{darkblue}{\frac{233}{144}}, & \frac{377}{233}, &\frac{610}{377} , & \color{darkblue}{\frac{987}{610} }, & \ldots \\ 1+\sqrt{5} & \approx & & & \color{darkblue}{3}, &&& \color{darkblue}{\frac{13}4}, &&& \color{darkblue}{\frac{55}{17}}, &&& \color{darkblue}{\frac{233}{72}}, &&& \color{darkblue}{\frac{987}{305}}, & \ldots \end{array} $$

This time all the convergents in the second list are matched by convergents in the first list. The number !!\frac{1+\sqrt5}{2}!! is notorious because it's the real number whose convergents converge the most slowly. I'm surprised that !!1+\sqrt5!! converges so much more quickly; I would not have expected the factor of 2 to change the situation so drastically.

I haven't thought about this at all yet, but it seems to me that a promising avenue would be to look at what Gosper's algorithm would do for the case !!x\mapsto 2x!! and see what simplifications can be done. This would probably produce some insight, and maybe a method for constructing a number !!\alpha!! so that all the convergents of !!2\alpha!! are twice those of !!\alpha!!.

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