  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 $\ell$.
