Abstract
Let $\mathcal{E}$ be a set of primes with density $\tau > 0$ in the set of primes. Write $\mathcal{A}$ for the set of positive integers composed solely of primes from $\mathcal{E}$. We discuss the distribution of the integers from $\mathcal{A}$ in short intervals, and whether for fixed $k \in \mathbb{Z}$ there are solutions to $m+k = p$ with $m \in \mathcal{A}$, where $p$ denotes a prime, or $m+k=n$ where $n$ has a large prime factor ($>n^{\xi}$ for $\xi > \tfrac{1}{2}$
Citation
Glyn Harman. Kaisa Matomäki. "Some problems of analytic number theory on arithmetic semigroups." Funct. Approx. Comment. Math. 38 (1) 21 - 39, January 2008. https://doi.org/10.7169/facm/1229624649
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