In the paper, we establish a generalized result of Heittokangas et al. (Complex Var Elliptic Equ 56(1–4):81–92, 2011) concerning the sharing of distinct periodic small functions with period c for meromorphic functions in \(\mathbb {C}^m\) . We demonstrate that if f(z) and \(f(z+c)\) share three distinct c-periodic small functions \(a_1(z)\) , \(a_2(z)\) and \(a_3(z)\) CM, where f(z) is a non-constant meromorphic function in \(\mathbb {C}^m\) , then either \(f(z) = f(z+c)\) or \(f(z) = f(z+2c)\) , with the latter case arising when \(\limsup \limits _{r\rightarrow \infty }\frac{\log T(r,f)}{r}>0\) . This extends the result of Heittokangas et al. (Complex Var. Elliptic Equ. 56(1–4):81–92, 2011) for finite order meromorphic function in \(\mathbb {C}\) to meromorphic functions in higher dimensions. Moreover, we show by plenty of examples that our results are best possible in certain senses.
