TITLE: Circles avoiding rational points of height $\le h$
QUESTION [10 upvotes]: 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!

REPLY [7 votes]: I'm going to interpret the question as:

What is the positive rational number of least height that cannot be written as the sum of $d$ squares of rational numbers of height $<h$?

An easier question might be:

What is the least height of a positive rational number that cannot be written as sum of $d$ squares of rational numbers of height $<h$?

There is a problem for $d=2$, which is some rational numbers can't be written as a sum of rational squares. I believe the smallest height ones are $3$ and $3/5$, so the circle of radius $\sqrt{3}$ and the sphere of radius $\sqrt{3/5}$ might be good answers to this version of  the question.
Thus from now on I'm going to assume we do have at least one rational point. For a lower bound, we may use the fact that $a/b$ is a sum of $d$ rational squares if and only if $ab$ is a sum of $d$ integer squares. This follows from the $4$-squares theorem if $d\geq 4$. If $d=2$ or $3$ we can look explicitly at the conditions for $ab$ to be a sum of squares, and note that they are clearly necessary conditions for $a/b$ to be a sum of squares. This lets us write $a/b$ as a sum of squares whose denominator is at most $b$, hence whose numerator is at most $\sqrt{ab}$. This gives a lower bound of $h$.
To get an upper bound close to $h$, let $r^2 =1/p$ for $p$ a prime number slightly larger than $h$ (and, if $d=2$, congruent to $1$ mod $4$, and if $d=3$, not congruent to $7$ mod $8$).  Then any rational points which sum to equal $1/p$ must have $p$ somewhere in the denominator of one of them, so must have height at least $p$, hence height $>h$.
So $r^2=1/p$ works quite well. You might find $r^2=a/p$, for $a$ slightly less than $p$ and $a$ a sum of two squares, more aesthetically pleasing.