A recent paper of Klurman and Mangerel proves that any completely multiplicative function $f: \mathbb{N} \to \mathbb{T}$ satisfies
$$\liminf_{n \to \infty} |f(n+1)-f(n)|=0.$$
See https://arxiv.org/abs/1707.07817. A stronger result for multiplicative functions is also available. Gowers' quantitative improvement of Szemeredi's theorem (as a black box) is one of three or so key ingredients.