Gap problems for integer-valued multiplicative functions

Motivated by questions about the typical sizes of gaps $|f(n+1)-f(n)|$ in the sequence $(f(n))_n$, where $f$ is an integer-valued multiplicative function, we investigate the set of solutions $$\{n \in \mathbb{N} : f(n+a) = f(n) + b\}, \quad ab \neq 0.$$ We formulate a conjecture classifying those multiplicative functions for which this set has logarithmic density zero, and prove that the conjectured classification is tight. Moreover, using techniques from additive combinatorics building on previous work of the author, we show how to reduce the classification problem to the study of "local power maps" modulo prime $\ell$, i.e., maps $g: \mathbb{N} \to \mathbb{Z}$ for which there is $0 \leq k_{\ell} < \ell-1$ such that $$g(n) \equiv n^{k_{\ell}} \pmod{\ell} \text{ for all } n \in \mathbb{N}.$$ We prove a partial result towards our classification conjecture by employing a strategy of N. Jones that uses Kummer theory to study local power maps modulo many primes $\ell$.