On Equal Consecutive Values of Multiplicative Functions

Let $f:\mathbb{N}\to \mathbb{C}$ be a multiplicative function for which $$\sum _{p:\mathrm{\mid }f(p)\mathrm{\mid }\mathrm{\ne }1}\frac{1}{p}=\mathrm{\infty }\mathrm{.}$$ We show under this condition alone that for any integer $h\mathrm{\ne }0$ the set $$\{n\in \mathbb{N}:f(n)=f(n+h)\mathrm{\ne }0\}$$ 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 $\mathrm{\mid }f{\mathrm{\mid }}^{it}$, 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.