  We prove that any positive rational number is the sum of distinct unit
fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same
conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any
$h\in\mathbb{Z}\backslash\{0\}$, provided a necessary congruence condition is
satisfied. We also prove that this is true for any subset of the primes of
relative positive density, provided a necessary congruence condition is
satisfied.
