Discrete uniform limit law for a sum of additive functions on shifted primes (Q1795413)

The authors study the following question. Let $\{f, g\}$ be a pair of strongly additive functions such that for each prime $p$ both$f(p)$ and $g(p)$ are in $\{0, 1 \}$ and let $L \ge 2$ be an integer. When is the sequence $(f(p+1) + g(p+2))_p$ uniformly distributed modulo $L$? Two obvious examples are the cases when $L=2$, $f$ and $g$ vanish on all the primes different from $3$ and $f(3) + g(3) = 1$ or $f$ and $g$ vanish on all the primes different from $5$ and $f(5) + g(5) = 2$. A consequence of the main theorem of the authors is that under a condition of the type $f(p)$ and $g(p)$ are $o(1/\log(p))$ on average, the obvious above-mentioned examples are the only possibilities. The main result is actually stated for families of strongly additive functions $(f_x, g_x)$ with values in $\{0, 1\}$ satisfying the condition \[ \sum_{x^{\gamma} \le p \le x} (f_x(p) +g_x(p))/p = o(1/\log x), \text{ for any } \gamma \in (0, 1). \] The proof uses the method of factorial moments.