Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\| = \min(\{x\}, 1 - \{x\})$, where $\lfloor x \rfloor$ is, as usual, the greatest integer $\le x$.


Let $x$ be a fixed irrational number and $\mu$ its irrationality measure, namely the infimum of the set of all positive exponents $s$ such that 


$$0 < \left|x - \frac{m}{n}\right| < \frac{1}{n^s}$$
for finitely many pairs $(m,n) \in \mathbb{Z} \times \mathbb{N}^+$. Assume that $\mu$ is finite (*added later: see N. Elkies' answer and the comments below*). Recall that $\mu \ge 2$ by (a corollary of) [Dirichlet's (approximation) theorem](http://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem). Then, given $s \in \{2\} \cup [2,\mu[$, define $G\_s$ as the set 


$$\{n \in \mathbb{N}^+: \|nx\| < n^{1-s}\},$$
and let $G\_s^{-} := \big\{n \in G\_s: \|nx\| = \{nx\}\big\}$ and $G\_s^+ := G\_s \setminus G\_s^{-}$. 


We have that $G\_s$ is infinite (by Dirichlet's theorem and the definition of $\mu$), so at least one of $G\_s^-$ or $G\_s^+$ is infinite too, and I'm tempted to claim that *each* of them *must* be infinite. I don't have a serious argument in support of this: My only point is that it would be weird, I believe, to observe a similar "asymmetry" in the diophantine approximations of $x$ (yet, I would be happy to hear that my expectation is completely wrong). This leads to the following:



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> > **Q1.** Is it true that $G\_s^-$ is infinite for each $s \in \{2\} \cup [2,\mu[$? **Q2.** If Q1 is well-established, would you kindly provide me with a reference?
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Some remarks: (i) If the answer to Q1 is yes, then it is as well true that $G\_s^+$ is infinite for each $s \in \{2\} \cup [2,\mu[$. (ii) I can prove that the answer to Q1 is positive if $s = 2$ (this follows from Dirichlet's theorem and some elementary properties of the simple continued fraction expansion of $x$). (iii) By (ii) and [Khintchine's theorem](http://en.wikipedia.org/wiki/Khinchin%27s_theorem_on_Diophantine_approximations#Khinchin.27s_theorem_and_extensions) (which yields that the irrationality measure of most real numbers is equal to $2$), the answer to Q1 is yes for almost all $x$.



Expected symmetry in the diophantine approximations of an irrational number
This can't be right in general. For example
$$
x = 0.101000001000000000000000001\ldots
 = \sum\_{k=0}^\infty 10^{-3^k}
$$
has plenty of lower approximants with exponent just under 3
(the partial sums) but no upper ones.


