Signed sums of prime reciprocals

Suppose we wish to show that at least one integer in $[n,n+\log n]$ contains a prime factor larger than, say, $\log^{100} n$ (thus breaking the square root barrier). We suppose this is not the case and obtain a contradiction.
It is then plausible that most integers in this interval contain about $\log^{1-o(1)} n$ or so factors between, say, $\log^{10} n$ and $\log^{100} n$. (Using the W-trick, one can probably eliminate primes much less than log n from consideration.) In particular, we get a set $p_1,\ldots,p_k$ of about $\log^{2-o(1)} n$ primes in $[\log^{10} n, \log^{100} n]$ which each divide an integer in $[n,n+\log n]$, which implies in particular that the n/p_i lies within $O(1/\log^9 n)$ of an integer. This implies that all of the 3^k signed sums
$\epsilon_1 n / p_1 + \ldots + \epsilon_k n / p_k$
with $\epsilon_i = -1,0,1$ lie within $O( 1 / \log^{7-o(1)} n )$ of an integer. Hopefully this sort of concentration leads to some sort of contradiction. For instance, it implies that $\epsilon_1/p_1 + \ldots + \epsilon_k/p_k$ cannot be used to approximate too many reciprocals 1/q too closely.