On Equal Consecutive Values of Multiplicative Functions

Abstract

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

20 pages; thanks to a suggestion by the anonymous referee, main result has been improved and the argument has been shortened