A reverse problem on arithmetic functions (Q1914034)

The author proves Kátai's conjecture: every multiplicative function \(f: \mathbb{N}\to \mathbb{C}\) with \[ |f(n) |=1 \text{ for } (n, B)=1, \qquad f(n+B)- f(n)= o(1), \quad (n,B)= 1, \quad n\to \infty \] for a suitable natural \(B\), is of the form \(f(n)= n^{i\tau} \chi_B (n)\), \((n, B)=1\) with real \(\tau\) and a Dirichlet character \(\chi_B \bmod B\). Special cases have been proved with similar methods by Kátai, Wirsing and Shao Pintsung.