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

TL;DR

This paper investigates the asymptotic behavior of a generating function related to the decimal expansion of powers of two, proposing a matrix representation to analyze upper bounds and dynamical properties.

Contribution

It introduces a novel matrix representation of the generating function and formulates an open problem on its upper bounds for powers of two.

Findings

Proposes a finite-state transfer operator model

Highlights the problem of bounding the generating function's growth

Provides a framework for future dynamical analysis

Abstract

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.