Hausdorff dimension of escaping sets of meromorphic functions II

A function which is transcendental and meromorphic in the plane has at least two singular values. On the one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either <![CDATA[ $2$ ]]> or <![CDATA[ $1/2$ ]]>. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in <![CDATA[ $[0,2]$ ]]> (cf. [M. Aspenberg and W. Cui. Hausdorff dimension of escaping sets of meromorphic functions. Trans. Amer. Math. Soc. 374(9) (2021), 6145-6178]). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than <![CDATA[ $4$ ]]>.