Sturmian word: Difference between revisions
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Stretch goal: calculate the Sturmian word for other kinds of definable real numbers, such as cubic roots.
The key difficulty is accurately calculating <math>floor(k\sqrt a) </math> for large <math>k</math>. Floating point arithmetic would lose precision. One can either do this simply by directly searching for some integer <math>a'</math> such that <math>a'^2 \leq k^2a < (a'+1)^2</math>, or by more trickly methods, such as first calculating the continued fraction representation of <math>\sqrt a</math>, then obtaining the convergent sequence to <math>\sqrt a</math>. Let <math>\frac mn </math> be a convergent to <math>\sqrt a</math>, such that <math>n \geq k</math>, then since the convergent sequence is the '''best rational approximant''' for denominators up to that point, we know for sure that, if we write out <math>\frac{0}{k}, \frac{1}{k}, \dots</math>, the sequence would stride right across the gap <math>(m/n, 2x - m/n)</math>. Thus, we can take the largest <math>l</math> such that <math>l/k \leq m/n</math>, and we would know for sure that <math>l = floor(k\sqrt a)</math>.
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Revision as of 23:11, 31 January 2024
![Task](http://static.miraheze.org/rosettacodewiki/thumb/b/ba/Rcode-button-task-crushed.png/64px-Rcode-button-task-crushed.png)
You are encouraged to solve this task according to the task description, using any language you may know.
A Sturmian word is a binary sequence, finite or infinite, that makes up the cutting sequence for a positive real number x, as shown in the picture.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Fibonacci_word_cutting_sequence.png/300px-Fibonacci_word_cutting_sequence.png)
The Sturmian word can be computed thus as an algorithm:
- If , then it is the inverse of the Sturmian word for . So we have reduced to the case of .
- Iterate over
- If is an integer, then the program terminates. Else, if , then the program outputs 0, else, it outputs 10.
The problem:
- Given a positive rational number , specified by two positive integers , output its entire Sturmian word.
- Given a quadratic real number , specified by three positive integers , where is not a perfect square, output the first letters of its Sturmian word when given a positive number .
(If the programming language can represent infinite data structures, then that works too.)
Stretch goal: calculate the Sturmian word for other kinds of definable real numbers, such as cubic roots.
The key difficulty is accurately calculating for large . Floating point arithmetic would lose precision. One can either do this simply by directly searching for some integer such that , or by more trickly methods, such as first calculating the continued fraction representation of , then obtaining the convergent sequence to . Let be a convergent to , such that , then since the convergent sequence is the best rational approximant for denominators up to that point, we know for sure that, if we write out , the sequence would stride right across the gap . Thus, we can take the largest such that , and we would know for sure that .