Irregular set and metric mean dimension with potential

Let $(X,f)$ be a dynamical system with the specification property and $\phi$ be a continuous function. In this paper, we consider the multifractal irregular set \begin{align*} I_{\varphi}=\left{x\in X:\lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^ix)\ $\text{does not exist}$\right} \end{align*} and show that this set is either empty or carries full Bowen upper and lower metric mean dimension with potential.