This is a follow-up to [Question 146635](https://mathoverflow.net/questions/146635/expected-symmetry-in-the-diophantine-approximations-of-an-irrational-number), namely *Expected symmetry in the diophantine approximations of an irrational number*, which I will refer to for notation and terminology used here without explanation.


Let $x$ be an irrational number and $\mu$ its irrationality measure. For each $s \in \{2\} \cup [2,\mu[$ set 


$$G\_s := \{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$ and then define $G\_s^- := \big\{n \in \mathbb{N}^+: \|nx\| = \{nx\}\big\}$ and $G\_s^+ := G\_s \setminus G\_s^-$ (recall that $\mu \ge 2$). It was already remarked in the other thread that $G\_s$ is infinite (by Dirichlet's approximation theorem and the definition itself of $\mu$), so at least one of $G\_s^-$ or $G\_s^+$ is infinite too. 


However, while $G\_2^-$ and $G\_2^+$ are *both* infinite, [Noam Elkies' example](https://mathoverflow.net/a/146668) proves that it can well happen that one of $G\_s^-$ or $G\_s^+$ is finite for all $s \in \; ]2,\mu[$. Moreover, the same example can be generalized to produce a *countably* infinite family of such examples, by considering Mahler numbers of the form $\sum\_{n=0}^\infty a^{-b^n}$, where $a$ and $b$ are integers with $a \ge 2$ and $b \ge 3$ (each of these numbers has irrationality measure equal to $b$; see [here](https://mathoverflow.net/questions/145484/on-the-irrationality-measure-of-sum-n-1-infty-a-bn) for a discussion on this point). As for the case $\mu = \infty$, we can consider Liouville numbers of the form $\sum\_{n=0}^\infty 2^{-(n!)^k}$ for $k = 1, 2, \ldots$


Having the above in mind, let us say that $x$ is a *balanced number* if both $G\_s^-$ and $G\_s^+$ are infinite for all $s \in \;]2, \mu[$, and *unbalanced* otherwise. As mentioned in the other thread, almost all numbers are balanced (just because almost all numbers have an irrationality measure equal to $2$). It seems then natural to ask the following:



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I'm pretty sure that the question of deciding whether or not a number is balanced has been stated and studied before, possibly in a more general form. If this is actually the case, then:



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> > **Q3.** Would you kindly provide me with a reference?
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Some further remarks: When $2 \le \mu < \infty$, a rough idea could be to consider a number $x$ of the form $2 \cdot \sum\_{n = v}^\infty 3^{-\lfloor \lambda \mu^n\rfloor}$, where $\lambda \in \mathbb{R}^+$ and $v$ is a sufficiently large integer (depending on $\lambda$ and $\mu$). It is known after Y. Bugeaud's *Diophantine approximations and Cantor sets* (Math. Ann.
**341**, No. 3 (2008), 677-684) that the irrationality measure of $x$ is $\mu$. However, it seems a little difficult to make Noam's reasoning work in this case too (but I would be rather happy to learn that I'm wrong). Furthermore, this would not cover the case when $\mu$ is infinite.



Numbers with balanced diophantine approximations
Yes, there are uncountably many examples (maybe I should have mentioned this variation with my previous answer). For example, $\sum\_{k=0}^\infty b\_k 10^{-3^k}$ with each $b\_k=1$ or $2$. There are uncountably many choices (cardinality of the continuum, even), and for each there are infinitely many lower approximants with exponent approaching 3 (the partial sums, as before), and no comparably good upper approximants.


