Approximating a real by a ratio of primes Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$.
I seek smallest primes $p$ and $q$ such that 
$$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$

Q. What upper bound $u(\epsilon)$ can be placed on $\max\{p,q\}=\max(q)$ as a function of $\epsilon$?

Examples.
$x=\sqrt{2}/2 \approx 0.707107$, $\epsilon=10^{-4}$, $y=(x+\epsilon)$, then
$$
x = \sqrt{2}/2 \approx 
0.\color{blue}{707}{\color{red}{1}}07 < \frac{p}{q}=\frac{1217}{1721}\approx 0.\color{blue}{707}147 < 0.\color{blue}{707}{\color{red}{2}}07
\approx (\sqrt{2}/2 + \epsilon) = y \;.
$$
For $\epsilon=10^{-5}$, 
$$
x = \sqrt{2}/2 \approx 
0.\color{blue}{7071}\color{red}{0}7 < \frac{p}{q}=\frac{3491}{4937}\approx 0.\color{blue}{7071}096 < 0.\color{blue}{7071}{\color{red}{1}}7
\approx (\sqrt{2}/2 + \epsilon) = y \;.
$$
I believe those are optimal prime ratios for those $x$ & $y$.
If so, $u(10^{-4}) \ge 1721$ and $u(10^{-5}) \ge 4973$;
and $u(10^{-6}) \ge 8597$ 
and $u(10^{-7}) \ge 38287$
(data not presented):
$$
\begin{array}{cccc}
\epsilon=10^{-4} & \epsilon=10^{-5} & \epsilon=10^{-6} & \epsilon=10^{-7}\\
u(\epsilon) \ge 1721 & u(\epsilon)\ge 4937 & u(\epsilon)\ge 8597 & u(\epsilon)\ge 38287
\end{array}
$$

(The above question is tangentially related to the earlier question,
"Visibility in a prime orchard.")
 A: Here is an idea which might help show that the smallest prime is not much larger than the smallest integer needed.
One of the processes I like to use is the mediant $\frac{a+c}{b+d}$ of two positive fractions $\frac ab$ and $\frac cd$ to approximate a real with a fraction of small denominator.  Suppose we use the process to approximate $x +\epsilon/2$, say.  At some point we will end with two ratios of integers of, say, $k$ decimal digits each, call them $\frac ab$ and $\frac cd$,
with $$x \lt \frac ab \lt x + \frac{\epsilon}{2} \lt \frac cd \lt x + \epsilon$$.
Then the mediant $\frac{a+c}{b+d}$ will have at most $k+1$ digits in numerator and denominator, and this can be tweaked to adjust the numerator and denominator to be primes without moving the ratio out of the interval of interest.  Probably one can tweak $\frac ab$ or $\frac cd$ similarly, but the point is that the prime representation won't need many more digits than the pure integer representation, and if $\epsilon$ is measured in negative powers of 10, I predict fewer decimal digits than that will be needed for the prime representation.
Gerhard "Caution: Much Vigorous Handwaving Nearby" Paseman, 2015.07.11
