Local density of numbers not divisible by small primes 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?
 A: Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently. For example, for $\theta=1/2$ the sifted set consists precisely of the primes in $(X^{1/2},X]$ whose density is $1/\log X$ (instead of $2e^{-\gamma}/\log X$) by the Prime Number Theorem.
Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula
$$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$
where $\omega:[1,\infty)\to[1/2,1]$ is the Buchstab function satisfying (cf. Corollary 3.1 after the theorem)
$$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$
Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.
A: It may be helpful to provide a combinatoric perspective for contrast.  It will raise some interesting questions.
Given $n$ equal to the $k$th prime represent the left hand side as a reduced fraction I call $Pk$.  For $k=1$ to $4$ we have the fractions 1/2, 1/3, 4/15, and 8/35.  Thus I would expect an interval containing numbers coprime to $210$ to have this density be 8/35 only if the length of the interval is a multiple of 35.  This quickly leads to the question: how fast does the denominator of $Pk$ grow?
A related statistic is the ratio $\omega(\prod (p-1))$, the number of distinct prime factors of the (unreduced) numerator of $Pk$; for $k\lt 46$ this quantity is lower bounded by $k/3$.  I do not know the asymptotic of this quantity, but to discuss the previous paragraph further let me conjecture that it stays above $k/3$.
If it does stay above $k/3$, this means that the reduced denominator of $Pk$ is (very roughly) at most $P^{2/3}$, where $P$ is the product of the first $k$ primes. So for such $P$ (and thus for $X$ not far from $P$), I would expect an interval of length about $P^{2/3}$ to exactly match the density of coprimes to P in the interval $(\log P, P) \approx (p_k,P)$.  If you are willing to accept some error, I would expect much smaller intervals to exist (of almost any length greater than say $2\log P$) to come close to the desired density. ( Of course, I am interested in how much the density can vary at that scale, which I believe can go from 0 to almost twice the average.) As remarked in another answer, if you fix $\theta \gt 0$ and ask ( as $X$ goes to infinity ) about density of numbers less than $X$ with no prime factor less than $X^\theta$, that density drops considerably.
Gerhard "It's A Matter Of Scale" Paseman, 2016.12.06.
