Since$$\prod_{p \leq n} \left(1-\frac{1}{p}\right) =\frac{ e^{-\gamma}+o(1)}{ \log n},$$ by Mertens theorem, the density of integers in $$(X^{\theta},X],$$ which aren't divisible by primes $$p \leq X^{\theta}$$ is $$\rho(X)\sim \frac{ e^{-\gamma}}{\theta \log n}.$$

How small a subinterval in this interval can inherit this density?

Since$$\prod_{p \leq n} \left(1-\frac{1}{p}\right) =\frac{ e^{-\gamma}+o(1)}{ \log n},$$ by Mertens theorem, the density of integers in $$(X^{\theta},X],$$ which aren't divisible by primes $$p \leq X^{\theta}$$ is $$\rho(X)\sim \frac{ e^{-\gamma}}{\theta \log X}.$$

How small a subinterval in this interval can inherit this density?