Upper Bounds for Digitwise Generating Functions of Powers of Two: A Problem and a Matrix Representation

This short note studies the asymptotic behavior of a generating function associated with the decimal expansion of $2^n$. Our aims are twofold: (i) to present a problem on the best possible upper bound for this behavior, and (ii) to introduce a matrix representation that is useful for its analysis. The representation corresponds to a finite-state transfer operator; analytic and dynamical aspects are not pursued here.