Mersenne almost primes

Question

I asked earlier whether it can be proved that infinitely many elements of $P_n$ for some positive value of $n$ (here $P_n$ refers to the set of numbers with at most $n$ prime divisors). There I received a positive answer in the form of a 1978 paper of Iwaniec which confirmed that suitable quadratic polynomials do in fact hit $P_2$ infinitely often.

My question now concerns the situation with Mersenne primes. What is the least value of $n$, if it is known to be finite, such that it is known that infinitely many Mersenne numbers $2^p - 1$ (where $p$ is a prime) hits $P_n$ infinitely often. In other words, there exist infinitely many primes $p$ such that $2^p - 1$ has at most $n$ prime factors.

Edit: in view of the relatively clear answer given by http://www.math.ucsd.edu/~asalehig/SG_AffineSieveExpandersOverviewMSRI.pdf, it seems that what I asked is an open problem without much chance of being resolved in the near future, so I ask this modified question:

Let $M(x) = \{2^p - 1 : p \leq x\}$ where $p$ as usual denotes a prime. Is it possible to find two functions $f(x), g(x)$ that tend to infinity with $f$ taking on values in the positive integers such that $$\displaystyle |M(x) \cap P_{f(x))}| \gg g(x)?$$

In other words, can it be shown that the set of Mersenne numbers with $p \leq x$ having relatively few prime factors (here $f$ is understood to tend to infinity slowly) tends to infinity (perhaps slowly) as $x$ tends to infinity?

Answer 1

This is a classical problem, and it remains open to show that there exist arbitrarily large Mersenne numbers with a bounded number of prime factors. The work of Bourgain, Gamburd, and Sarnak on the affine sieve may be seen as a generalization of this kind of question, but their work does not say anything for Mersenne numbers. An exposition of this work may be found in Kowalski's Bourbaki article http://arxiv.org/abs/1012.2793 ; also see these lecture notes of Golsefidy http://www.math.ucsd.edu/~asalehig/SG_AffineSieveExpandersOverviewMSRI.pdf which explicitly mentiones the connection with Mersenne numbers.