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 $r$?
Here is a python function bsa(r,n) that returns a list of all best $S$-approximations to $r$ with denominator at most $n^2$.
lst=[(floor(r+.5),1)] for den in range(2,n+1): dens,last=den*den,lst[-1]; num=floor(r*dens+.5) if abs(r-num/dens)<abs(r-last/last): lst.append( (num,dens) ) return lst
(Here $S$ is the squares.) Running it on 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?