Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes.
Problem 2: Find a (not extremely artificial) set $A$ of integers so that for every $n$, $ |A\cap [n]| \le n^{0.499}$, where you can prove that
$ \sum \{\mu(k): k \le n, k \in A\} = o(|A \cap [n]).$
###Variation
For problem 1 it makes sense to ask not just about infinitely many primes but about a result regarding the density of primes which is of the full strength of the prime number theorem. (In fact, I thought about Problem 2 as a weak form of problem 1 but this is not the case the way I formulated it.)
Problem 3: Find a (not extremely artificial) set $A$ of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that the density of primes in $A$ in the interval $[n]$ is $1/\log n+o(1)$.
Perhaps the best way to formulate and think about Problem 3 is in terms of orthogonality with the Van Mangold function.
Motivations:
This question is motivated by various recent results on Mobius randomness and infinitude of primes in various exotic sets, and also on this question: Why so difficult to prove infinitely many restricted primes?.
(The set of $p_{n^5}$ is extremely artificial and I suppose that a set that can be ordered in a sequence (not necessarily increasing) such that $a_n$ can be provably computed in $poly (\log |a_n|)$ steps is not extremely artificial.)
Examples:
A very natural example is an interval of the form $[n,n+t]$ and here indeed the best known absolute results is when $t=n^{0.535..}$. RH allows $t=\sqrt n \log n$ and it looks that here the $n^{1/2}$ is a viable barrier. (Here I base the info on the paper: A Survey of Results on Primes in Short Intervals by Cem Yalçın YILDIRIM.)
There is a result by Friedlander and Iwaniec that there are infinitely many primes of the form $a^2+b^4$. Here the density is above the square-root barrier, but I don't know if there are any insights regarding improvement to, say, $a^3+b^7$.
There is a result by Elkies about the infinitude of supersingular primes. Those primes are conjectured to be of density $n^{1/2}$ among the first $n$ numbers but provably it is only known that they are of density less or equal $n^{3/4}$ I don't know if there are Elkies-like results that can lead (provably or conjecturally) to sparser sets of primes.
There is a "PNT for majority functions" result of Bourgain that implies that there are infinitely many m-digits primes with more than a fraction $c$ of their binary digits are ones, for some $c=1/2+m^{-\rho}$, for some $\rho<1$ Having this for $c=0.9$ will cross the square root barrier. (Update based on Christian's answer: )Eric Naslund used the results about primes in intervals to prove it for $c=0.737..$.
