Is the sequence \sqrt{p}-\lfloor\sqrt{p}\rfloor\sqrt{p}-\lfloor\sqrt{p}\rfloor , pp running over the primes , dense in [0,1][0,1]?

Another formulation of the title :

If we have real numbers $a,b$ with $0, can we always find a prime $p$ such that $a<\sqrt{p}-\lfloor\sqrt{p}\rfloor holds ?

The fractional parts of the numbers $\sqrt{2}\cdot n$ , $n$ running over the positive integers, are known to be even equidistributed modulo $1$, so we can always find a natural number $n$ with $a<\sqrt{n}-\lfloor\sqrt{n}\rfloor for $a,b$ as above, but what is the situation if we restrict to the primes ?

The question is whether the fractional parts $\{\sqrt{p}\}$, with $p$ running over primes, are dense in $[0,1]$. One can show the stronger result that the fractional parts are equidistributed modulo 1.
for each fixed integer $h \neq 0$, the implied constant possibly depending on $h$. Here $\Lambda$ is the von Mangoldt function and $e(x) = e^{2\pi i x}$. This exponential sum can be bounded using Vaughan's identity and some straightforward exponential sum estimates.