A month ago it occurred to me to try and figure out how well the rationals approximated the irrationals. I am aware of the irrationality measure but I wondered whether there might be a method for determining how well it was approximated by an infinite number of rationals. The function that resulted from my analysis was the following:

, let denote the optimal rational approximants of :

Using the language of functions rather than sequences we may define:

It’s not known to me whether is convergent for any irrational . But, I believe that the function is not bounded anywhere:

For any open interval

While I haven’t had any success with this particular question so far I have managed to show that for uncountably many irrationals:

Let’s define the uncountable sets:

It isn’t too difficult to show that:

Now, using (5) it follows that:

p.s. This function has probably been studied before but after several unsuccessful google searches I decided to give it this name.

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