We prove that the Hausdorff dimension stays the same no matter how slowly the function φ grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.
Funding
GGR, MH, and NS were supported by The Australian Research Council discovery project grant number 200100994. HT was supported by the JSPS KAKENHI 19K21835, 20H01811.