Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ Note that for distinct $\frac{a}{q}, \frac{a'}{q'}\in A$, we have $$\left\lVert \frac{a}{q}  \frac{a'}{q'}\right\rVert\ge\frac{1}{qq'}\ge\frac{1}{X^2},\quad\text{where}\quad\lVert\beta\rVert := \min_{n\in\mathbb{Z}} \beta  n.$$ It follows that in any interval of length $X^{2}$ in $[0, 1]$, there are at most $O(1)$ elements of $A$, and this is clearly the best result possible, since $A\gg X^2$. Is it possible to get better results for $B$, since we have that $B\asymp X^2 / \log X$ by the prime number theorem. In particular, is it possible to get a bound of $o(\log X)$ for the number of elements of $B$ in an interval of length $$B^{1}\ll X^{2}\log X?$$ Note that the inclusion $B\subseteq A$ gives the trivial bound $O(\log X)$.

$\begingroup$ With the current definition, we have $B = \{ A\}$, are you sure that is what you want? $\endgroup$ – Dirk Jul 10 '18 at 10:51

$\begingroup$ @Dirk, I'm guessing OP has forgotten to tell us that $p$ is restricted to being a prime. $\endgroup$ – Gerry Myerson Jul 10 '18 at 12:35

1$\begingroup$ I've corrected these typos. The inequality was originally the wrong way around. $\endgroup$ – Mayank Pandey Jul 10 '18 at 19:50

1$\begingroup$ I wonder whether the paper Harman, Glyn, Numbers badly approximable by fractions with prime denominator, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 1–5, MR1329453 (96c:11080) is relevant here. $\endgroup$ – Gerry Myerson Jul 11 '18 at 0:45

2$\begingroup$ This type of inequality on fractions is the starting point of the large sieve inequality. In case that you are actually interested in applications with the large sieve: Wolke observed that the sieve estimates with prime denominators can almost save the log factor: Wolke, D. On the large sieve with primes. Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247. mathscinet.ams.org/mathscinetgetitem?mr=291121 $\endgroup$ – Christian Elsholtz Jul 17 '18 at 21:43
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Wolke (On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247, MathSciNet MR0291121 (45 #215)) has worked on this question. His motivation was Gallagher's approach to the large sieve. If I remember correctly, his result was almost saving a factor $\log x$, which is what one would expect.