Let \(\langle d_1(x),d_2(x),d_3(x),\dots \rangle \) denote the Pierce expansion of an irrational number \(x\in (0,1]\) . Shallit (1986) proved that, for Lebesgue almost every \(x\in (0,1]\) , the growth rate of \(\{\log d_n(x)\}_{n\in \mathbb N}\) is linear. Given a function \(\varphi :{\mathbb {N}}\rightarrow {\mathbb {R}}^{+}\) with \(\varphi (n)\rightarrow \infty \) as \(n\rightarrow \infty \) , this paper studies the limit behavior of \(\{\log d_n(x)\}_{n\in \mathbb N}\) relative to \(\varphi \) from the perspective of multifractal analysis. Assuming that the limit \(\rho :=\lim _{n\rightarrow \infty }\frac{\varphi (n)}{\log n}\) exists, we determine the Hausdorff dimension of the set of all \(x\in (0,1]\) for which the lower (or upper) limit of \(\frac{\log d_n(x)}{\varphi (n)}\) equals one, according to the values of \(\rho \) .
