User:A/Markov constant: Difference between revisions

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We can investigate further by bringing in the broader idea of '''Markov constants''' from Diophantine approximation theory, which provide a method to measure how difficult it is to approximate a certain irrational number with rationals. Every irrational number has a Markov constant <math>M(x)</math> which is either infinite or a positive real number. <math>M(x)</math> is finite iff <math>x</math>'s simple continued fraction is bounded.

If <math>y = \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math>, then <math>M(x) = M(y)</math> and <math>x</math> and <math>y</math> are equivalent in Diophantine approximation. Transformations of the form <math>x \mapsto \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math> form a group, which if my Wikipedia understanding of the modular group is accurate, is generated by the transformations <math>x \mapsto -x</math>, <math>x \mapsto \frac{~~x +~~ 1}{x}</math>, and <math>x \mapsto \frac{1}{x - 1}</math>.

If <math>y = \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math>, then <math>M(x) = M(y)</math> and <math>x</math> and <math>y</math> are equivalent in Diophantine approximation. Transformations of the form <math>x \mapsto \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math> form a group, which if my Wikipedia understanding of the modular group is accurate, is generated by the four transformations <math>x \mapsto -x</math>, <math>x \mapsto 1 \pm \frac{1}{x}</math>, and <math>x \mapsto \frac{1}{x \pm 1}</math>.

Smaller <math>M(x)</math> have slower convergence in their continued fractions. We can treat <math>M(x)</math> as a somewhat sideways measure of "consonance," although there are no guarantees that it bears any relationship with actual consonance. The irrational numbers with the smallest Markov constants, and are therefore "most dissonant" by this measure, are:

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 We can investigate further by bringing in the broader idea of '''Markov constants''' from Diophantine approximation theory, which provide a method to measure how difficult it is to approximate a certain irrational number with rationals. Every irrational number has a Markov constant <math>M(x)</math> which is either infinite or a positive real number. <math>M(x)</math> is finite iff <math>x</math>'s simple continued fraction is bounded.
-If <math>y = \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math>, then <math>M(x) = M(y)</math> and <math>x</math> and <math>y</math> are equivalent in Diophantine approximation. Transformations of the form <math>x \mapsto \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math> form a group, which if my Wikipedia understanding of the modular group is accurate, is generated by the transformations <math>x \mapsto -x</math>, <math>x \mapsto \frac{x + 1}{x}</math>, and <math>x \mapsto \frac{1}{x - 1}</math>.
+If <math>y = \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math>, then <math>M(x) = M(y)</math> and <math>x</math> and <math>y</math> are equivalent in Diophantine approximation. Transformations of the form <math>x \mapsto \frac{ax + b}{cx + d}</math> with integers <math>ad - bc = \pm 1</math> form a group, which if my Wikipedia understanding of the modular group is accurate, is generated by the four transformations <math>x \mapsto -x</math>, <math>x \mapsto 1 \pm \frac{1}{x}</math>, and <math>x \mapsto \frac{1}{x \pm 1}</math>.
 Smaller <math>M(x)</math> have slower convergence in their continued fractions. We can treat <math>M(x)</math> as a somewhat sideways measure of "consonance," although there are no guarantees that it bears any relationship with actual consonance. The irrational numbers with the smallest Markov constants, and are therefore "most dissonant" by this measure, are: