Is there any set $X$ which is a density 0 subset of $N^*$ and we already know that there are infinitely many primes in it, beside examples which come from $x^2+y^4$(or its proof)?

Problem1: In particular, is it already proved that there exist $c>1$, s.t. $A_c=\{n\in\mathbb{N}^*| \exists k\in\mathbb{N}^* , n=[k^c]\}$ contains infinitely many primes?

If this problem can not be solved by existing methods, can the following one be solved by existing methods?

Problem2: for all $c>1$, $\exists d(c), c>d(c)>0$, s.t. $A_c=\{n\in\mathbb{N}^*| \exists k\in\mathbb{N}^* , n=[k^{e}], e\in (c-d(c),c+d(c))\}$ contains infinitely many primes.

This is only a naive question, thanks in advance.