If I understand correctly, in the following thread

Are There Primes of Every Hamming Weight?

two users of the site claim that it has been already proven that, for every sufficiently large $n \in \mathbb{N}$, there exist primes numbers with Hamming weight equal to $n$. Their claim is apparently supported by Theorem 1.2 of the paper "Primes with an average sum of digits" by M. Drmota, C. Mauduit, and J. Rivat.

Do you know if there is a text out there in which the deduction of the existence of primes with Hamming weight $n$ from the said theorem by Drmota, Mauduit, and Rivat is established in a thorough manner? Like other users of the site (see the sections of comments in the aforementioned thread), I am not totally sure of the veracity of such a claim. In case that you believe that there is no author out there that has dealt with this topic in detail but you consider that you've gotten the idea of the proof, would you be so kind as to explain it below as though I were a five-year old?

Thanks in advance for your help!

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