How Symmetric is Diophantine Approximation using Fractions with Square Denominators? Let $S$ be an infinite set of positive integers. 
Let us say that a  "best S-approximation" to a real irrational $r$ is a rational number
$p/q$, with $p$ and $q$ integers and $q \in S$, such that for any integer $m\in S$ with $1 \le m<q$, and any integer $n$, it holds that $|p/q-r|<|n/m-r|$. 
In the case that $S$ is the positive integers, it is well known that for any  $r$ there are infinitely many best $S$-approximations $p/q$ which are less than $r$ and infinitely many that are greater than $r$. This is proved for example in Khintchin's book "Continued Fractions." 
What if $S$ is the set of squares of integers? Is it known that for every irrational $r$ there are infinitely many best $S$-approximations less than $r$ and infinitely many greater than $r$? 
Experimentation with numerous commonplace irrationals gives the impression that the ratio of the number of under-estimates to the number of over-estimates tends to 1. Could this be true for every irrational $r$? Is this known?
 A: I believe that a natural version of the problem is finding how good are rational approximations of the form $p/q^k$ to a given irrational number $\theta$. This classical problem goes back to H. Heilbronn [Quart. J. Math. (Oxford) 19 (1948) 249--256] and I. Danicic [Mathematika 5 (1958) 30--37] and the best known results are due to C. Hooley [in: Analytic number theory, Vol. 2 (Allerton Park, IL, 1995) 471–-486,
Progr. Math. 139 Birkhäuser Boston, Boston, MA, 1996] and G. Harman [Glasgow Math. J. 38 (1996) 299–308]. They show that there are infinitely many coprime pairs $p,q$ with the property
$$
\left|\theta-\frac p{q^k}\right|<\frac1{q^{k+\rho_k-\varepsilon}}
$$
where $\rho_2=\frac25$ and $\rho_k=1/(3\cdot 2^{k-2}-1)$ if $k\geq 3$.
A: First of all, I think when you say "any real" you mean "any real irrational." Second, you can find a real irrational such that all the even convergents, and none of the odd convergents, have square denominator. I think such an irrational would have infinitely many best $S$-approximations below but none above. 
EDIT: Glyn Harman wrote a series of 4 papers, Metric Diophantine approximation with two restricted variables, which may have some relevance here. 
