User:Akselai/RES comma: Difference between revisions
Created page with "'''Recursive equal step (RES) commas''' are a sequence of commas {''a''<sub>1</sub>, ''a''<sub>2</sub>, ...} generated by an initial interval ''a''<sub>0</sub> and a set o..." |
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In general, the more elements ''S'' contains, the smaller the comma after a fixed number of steps. The rate of diminishing of RES commas is an | In general, the more elements ''S'' contains, the smaller the comma after a fixed number of steps. The monzos coordinates appear to grow at the rate of factorial (or ''x''<sup>''x''</sup>). However, the rate of diminishing of RES commas is an unsolved problem. Nevertheless, we have an absolute upper bound of 2<sup>-''n''</sup>''a''<sub>0</sub>, because the distance from an interval to two equal steps cannot exceed half the distance between the two equal steps. | ||
= Miscellaneous = | = Miscellaneous = | ||
It is possible for an RES comma to be 1/1. Consider a modified version of the above example. We have ''a'' = 5/4, but this time ''S'' = {[-120 1 51⟩, 3/2, 2/1}. We get ''a''<sub>1</sub> = 128/125 and ''a''<sub>2</sub> = [-120 1 51⟩ again. But in the case of ''a''<sub>3</sub>, since ''a''<sub>2</sub> is already in the set of intervals, no other interval in the set is going to be closer than a power of ''a''<sub>2</sub>, so ''a''<sub>3</sub> = ''a''<sub>2</sub>/''a''<sub>2</sub> = 1/1. Since all the powers of 1/1 are 1/1, the closest interval is just the smallest interval in the set, so ''a''<sub>4</sub> = [-120 1 51⟩, ''a''<sub>5</sub> = 1/1, and so on. | It is possible for an RES comma to be 1/1. Consider a modified version of the above example. We have ''a'' = 5/4, but this time ''S'' = {[-120 1 51⟩, 3/2, 2/1}. We get ''a''<sub>1</sub> = 128/125 and ''a''<sub>2</sub> = [-120 1 51⟩ again. But in the case of ''a''<sub>3</sub>, since ''a''<sub>2</sub> is already in the set of intervals, no other interval in the set is going to be closer than a power of ''a''<sub>2</sub>, so ''a''<sub>3</sub> = ''a''<sub>2</sub>/''a''<sub>2</sub> = 1/1. Since all the powers of 1/1 are 1/1, the closest interval is just the smallest interval in the set, so ''a''<sub>4</sub> = [-120 1 51⟩, ''a''<sub>5</sub> = 1/1, and so on. | ||