  Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$
\sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this
condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} :
f(n) = f(n+h) \neq 0\} $$ has logarithmic density 0. We also prove a converse
result, along with an application to the Fourier coefficients of holomorphic
cusp forms.
  The proof involves analysing the value distribution of $f$ using the
compositions $|f|^{it}$, relying crucially on various applications of Tao's
theorem on logarithmically-averaged correlations of non-pretentious
multiplicative functions. Further key inputs arise from the inverse theory of
sumsets in continuous additive combinatorics.
