Asymptotic Analysis of Regular Sequences

TL;DR

This work develops a unified asymptotic framework for the summatory function of $q$-regular sequences, showing it decomposes into a finite sum of growth terms driven by eigenvalues of the sum of linear representation matrices, with periodic fluctuations computable via residues of the associated Dirichlet series. The approach blends Mellin--Perron summation with a pseudo-Tauberian argument to rigorously justify the fluctuations and to handle convergence, yielding precise Fourier coefficients that describe the fluctuations. The authors apply the theory to transducer-based sequences, esthetic numbers, and Pascal's rhombus, obtaining sharp, previously unknown formulas and enabling algorithmic computation of fluctuations. Overall, the paper provides a comprehensive, computable, and highly general method for the asymptotics of regular sequences, with broad applications in combinatorics, number theory, and computer science.

Abstract

In this article, $q$-regular sequences in the sense of Allouche and Shallit are analysed asymptotically. It is shown that the summatory function of a regular sequence can asymptotically be decomposed as a finite sum of periodic fluctuations multiplied by a scaling factor. Each of these terms corresponds to an eigenvalue of the sum of matrices of a linear representation of the sequence; only the eigenvalues of absolute value larger than the joint spectral radius of the matrices contribute terms which grow faster than the error term. The paper has a particular focus on the Fourier coefficients of the periodic fluctuations: They are expressed as residues of the corresponding Dirichlet generating function. This makes it possible to compute them in an efficient way. The asymptotic analysis deals with Mellin--Perron summations and uses two arguments to overcome convergence issues, namely Hölder regularity of the fluctuations together with a pseudo-Tauberian argument. Apart from the very general result, three examples are discussed in more detail: sequences defined as the sum of outputs written by a transducer when reading a $q$-ary expansion of the input; the amount of esthetic numbers in the first~$N$ natural numbers; and the number of odd entries in the rows of Pascal's rhombus. For these examples, very precise asymptotic formulæ are presented. In the latter two examples, prior to this analysis only rough estimates were known.

Figures (5)

• Figure 9.1: Automaton $\mathcal{A}$ recognizing esthetic numbers.
• Figure 9.2: Fluctuation in the main term of the asymptotic expansion of $X(N)$ for $q=4$. The figure shows $\f{\Phi_1}{u}$ (red) approximated by its trigonometric polynomial of degree $1999$ as well as $X(4^u) / N^{u(\log_4(\sqrt{5} + 1) - \frac{1}{2})}$ (blue).
• Figure 10.1: Pascal's rhombus modulo $2$.
• Figure 10.2: Fluctuation in the main term of the asymptotic expansion of $X(N)$. The figure shows $\f{\Phi}{u}$ (red) approximated by its trigonometric polynomial of degree $1999$ as well as $X(2^u) / 2^{u\gamma}$ (blue).
• Figure 12.1: Maps in the proof of Theorem \ref{['theorem:contribution-of-eigenspace']}.

Theorems & Definitions (61)

• Example 3.1
• Remark 3.2
• Theorem A: User-friendly All-In-One Theorem
• Example 3.3: Continuation of Example \ref{['example:binary-sum-of-digits']}
• Remark 5.1
• Theorem B
• Theorem C
• Remark 6.1
• Theorem D
• Remark 6.2