The Rocquiasami function

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:

\forall \alpha \in \mathbb{R}_+ \setminus \mathbb{Q} , let \{\alpha_q\}_{q=1}^\infty \subset \mathbb{Q} denote the optimal rational approximants of \alpha:

\forall q \in \mathbb{N}, |\alpha_q -\alpha| = \min_{n \in \mathbb{N}}|\frac{n}{q}-\alpha| \quad (1)

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

\begin{cases} f: \mathbb{R}_+ \setminus \mathbb{Q} \rightarrow\overline{\mathbb{R}} \\ f(\alpha) = \sum_{q=1}^{\infty} |\alpha_q-\alpha| \end{cases} \quad (2)

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

For any open interval I \subset \mathbb{R_+},\forall M \in \mathbb{R_+} \exists \alpha \in I, f(\alpha) > M

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

Let’s define the uncountable sets:

M = [0,1] \setminus {{\mathbb{Q} \cap [0,1]}}  \quad (3)

S = \{\frac{1}{2}+\frac{m}{2^n} : m \in M\}_{n=5}^\infty \quad (4)

It isn’t too difficult to show that:

\forall \alpha \in S, f(\alpha) \geq |1-\alpha|+|\frac{1}{2}-\alpha|+|\frac{2}{3}-\alpha| \quad (5)

Now, using (5) it follows that:

\forall \alpha \in S, f(\alpha) > \alpha \quad (6)

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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