Signed sums of prime reciprocals

From Polymath1Wiki

One approach to finding non-smooth numbers (which, assuming a factoring oracle, would lead to progress on the finding primes problem) is as follows.

Suppose we wish to show that at least one integer in [n,n + logn] contains a prime factor larger than, say, log¹⁰⁰n (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, log¹⁰n and log¹⁰⁰n. (Using the W-trick, one can probably eliminate primes much less than log n from consideration.) In particular, we get a set of about log^{2 − o(1)}n primes in [log¹⁰n,log¹⁰⁰n] which each divide an integer in [n,n + logn], which implies in particular that the n/p_i lies within O(1 / log⁹n) of an integer. This implies that all of the 3^k signed sums

with ε_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 cannot be used to approximate too many reciprocals 1/q too closely.