Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$) of radius $r < 1$ avoid all rational points of height $\le h$?

A *rational point* is a point all of whose coordinates are rational numbers.
A rational number $x= a/b$ in lowest terms
(i.e., gcd$(a,b)=1$) has *height*
$\max \lbrace |a|,|b| \rbrace$.
A *rational point
of height $h$* is a rational point all of whose coordinates are of height $\le h$.

For example, let $h=10$, and $r=\frac{7}{11}$
^{[Thanks for corrections by NAME_IN_CAPS and Noam Elkies]}.
Then $C(\frac{7}{11})$,
unless I am mistaken, avoids
all rational points of height $\le 10$, despite some near misses:

Again fix $h=10$, but for $r=\frac{1}{\sqrt{2}}$, $C(r)$ passes through $$\lbrace{ (\frac{1}{10},\frac{7}{10}), (\frac{1}{2},\frac{1}{2}), (\frac{7}{10},\frac{1}{10}) }\rbrace \;:$$

Perhaps I should phrase the question in the obverse sense:

Q'. Which origin-centered circles $C(r)$ (or spheres in dimension $d$) of radius $r < 1$ include at least one rational point of height $\le h$?

So I am seeking $r$ as a function of $h$. I am unfamiliar with this type of reasoning, but I seek circles that avoid low-height rational points. Thanks!