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For integers $a$ and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by $a$and $b$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. The action on $\mathbb{T}$ by$T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if$a$ and $b$ are multiplicatively independent, then the only $\times a,\times b$invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is theLebesgue measure. However, whether there exists a nontrivial $\times a,\timesb$ invariant and ergodic measure is not known. In this paper, we study theempirical measures of $x\in\mathbb{T}$ with respect to the $\times a,\times b$action and show that the set of $x$ such that the empirical measures of $x$ donot converge to any measure has Hausdorff dimension $1$ and the set of $x$ suchthat the empirical measures can approach a nontrivial $\times a,\times b$invariant measure has Hausdorff dimension zero. Furthermore, we obtain someequidistribution result about the $\times a,\times b$ orbit of $x$ in thecomplement of a set of Hausdorff dimension zero.

$\times a$ and $\times b$ empirical measures, the irregular set and entropy