Let \(\{a_n(x)\}_{n\ge 1}\) be the sequence of digits of \(x\in (0,1)\) in infinite iterated function systems with polynomial decay of the derivative. The study of the weighted products of multiple digits with some given growth rate \(\varphi (n)\) originated from the discussion of the improvability of Dirichlet’s theorem. Let \(\Lambda \) be the set of points whose digits are non-decreasing in such iterated function systems. For any \(0\le \alpha \le \beta \le +\infty \) and the weight vector \({\textbf{t}}=(t_0,\cdots ,t_m)\in {R}_{\ge 0}^{m+1},\) define \(\begin{aligned} E^{{\textbf{t}}}(\alpha ,\beta ):=\Bigg \{x\in (0,1):\ &\liminf \limits _{n\rightarrow \infty }\frac{\log \left( a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x)\right) }{\log n}=\alpha ,\\&\limsup \limits _{n\rightarrow \infty }\frac{\log \left( a_n^{t_0}(x)\cdots a_{n+m}^{t_m}(x)\right) }{\log n}=\beta \ \Bigg \}. \end{aligned}\) In this note, we determine the Hausdorff dimensions of \(E^{{\textbf{t}}}(\alpha ,\beta )\) and the intersection of sets \(E^{{\textbf{t}}}(\alpha ,\beta )\bigcap \Lambda .\)
