Let \(b>1,\ 1\le p\le b-1\) be two integers and \(\overline{\Sigma }_p\subset \{0,1,\cdots ,b-1\}\) be the alphabet with p elements. The run-length function \(R_{b}^p(x,n)\) is defined to be the longest run of the digits \(x_i\) in \(\overline{\Sigma }_p,\) where \(x_i\ (1\le i\le n)\) are the first n digits in the b-ary expansion of \(x\in [0,1).\) There are already theorems concerning the metric properties and the corresponding multifractal analysis related to the function \(R_{b}^p(x,n)\) in one integer base expansions at a time. For any \(0\le \alpha \le \infty \) , in this paper, we will show that the set of points x for which \(\alpha \) is an accumulation point of the sequence \(\left\{ \frac{R_{b}^p(x,n)}{\phi (n)}\right\} _{n\ge 1}\) as \(n\rightarrow \infty \) enjoy the so-called large intersection property under some constraints of the function \(\phi \) . As a result, we obtain that the Hausdorff dimension is preserved when such sets defined are countably intersected. This also allows us to obtain the sets of points for which all the limiting sequences of \(\left\{ \frac{R_{b}^p(x,n)}{\phi (n)}\right\} _{n\ge 1}\) in all integer base expansions fail to exist simultaneously, has full Hausdorff dimension.
