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

Hideaki Noda

TL;DR

This work addresses the problem of bounding the asymptotic growth of the weighted digit-generating function for the last $n$ decimal digits of $2^n$ by introducing a sparse, finite-state matrix framework. It derives an exact matrix-factorization for the weighted count over the digit set using matrices $M_n^{[h]}$ and residue-vectors, and defines the associated finite-state transfer operator $T_m^{[h]}$ to recast the problem in linear-algebra terms. A general upper bound is obtained via the 1-norm, showing $\limsup_{n\to\infty} \Psi_n(h) \le \log\left( \frac{\max\{E,O\}}{5} \right)$, with equality in the symmetric case $E=O$ yielding $\lim_{n\to\infty} \Psi_n(h)= \log\left( \frac{1}{10} \sum_{j=0}^9 h(j) \right)$. The paper also states an open problem for the best possible bound and discusses conditional implications for linear digit-sum growth under that bound via Chernoff-type arguments and large deviations, highlighting the utility of the matrix viewpoint for digit-statistics of powers of two.

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.

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

TL;DR

This work addresses the problem of bounding the asymptotic growth of the weighted digit-generating function for the last decimal digits of by introducing a sparse, finite-state matrix framework. It derives an exact matrix-factorization for the weighted count over the digit set using matrices and residue-vectors, and defines the associated finite-state transfer operator to recast the problem in linear-algebra terms. A general upper bound is obtained via the 1-norm, showing , with equality in the symmetric case yielding . The paper also states an open problem for the best possible bound and discusses conditional implications for linear digit-sum growth under that bound via Chernoff-type arguments and large deviations, highlighting the utility of the matrix viewpoint for digit-statistics of powers of two.

Abstract

This short note studies the asymptotic behavior of a generating function associated with the decimal expansion of . 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.
Paper Structure (6 sections, 7 theorems, 47 equations, 1 figure)

This paper contains 6 sections, 7 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Basic Properties
  3. Matrix Representation
  4. An Asymptotic Upper Bound for Psi_n(h)
  5. Conditional Implication under an Upper-Bound Assumption
  6. Numerical Computation

Key Result

Lemma 1

For $m\ge 1$ the matrix $M_m^{[h]}\in\mathbb{C}^{2^{m-1}\times 2^{m}}$ defined by eq:defMn, the sum of the entries in each column $t$ satisfies

Figures (1)

  • Figure 1: Numerical verification of the validity of Problem \ref{['prob:mgf']} and comparison of three possible upper-bound functions.

Theorems & Definitions (15)

  • Lemma 1: Parity structure of column sums
  • proof
  • Lemma 2: Four solutions to the compatibility congruence
  • proof
  • Lemma 3: A recursion by adding the first digit
  • proof
  • Proposition 1: Matrix representation
  • proof
  • Remark 1: Finite-state preimage transfer operator
  • Proposition 2: General upper bound using the 1-norm
  • ...and 5 more