Palindrome complexity versus factor complexity

Abstract: Let ${\bf x} = (a_i)<em>{i \geq 0}$ be an infinite word over a finite alphabet $Σ$. Let $ρ(n)$ be the factor complexity function for $\bf x$ and ${\rm Pal}(n)$ be the palindrome complexity function for $\bf x$. We give a new relationship between these two quantities; namely, if $\bf x$ is not ultimately periodic, then $$\lim</em>{n \rightarrow \infty} {{ {\rm Pal} (n) \log ({\rm Pal} (n) + 1)} \over {ρ(n)}} = 0.$$ Furthermore, we prove that the numerator in this result is essentially optimal.

• The paper establishes that for non-ultimately periodic infinite words, the ratio (Pal(n) log(Pal(n)+1))/ρ(n) asymptotically converges to zero.
• It employs rigorous combinatorial constructions and case analyses of exponential and subexponential growth to link palindrome and factor complexities.
• The study implies that palindromic factors, even when logarithmically amplified, are asymptotically negligible compared to overall factor complexity, impacting algorithm design.

Palindrome Complexity Versus Factor Complexity: An Analytical Summary

Introduction

This paper investigates the quantitative relationship between two classical combinatorial invariants of infinite words over finite alphabets: the factor complexity $\rho(n)$ and the palindrome complexity $\Pal(n)$. The factor complexity counts the number of distinct length-$n$ factors (subwords) in an infinite word ${\bf x}$, while the palindrome complexity counts the number of length-$n$ palindromic factors. While both measures have been studied extensively, their interdependence remains less understood. The primary contribution of the paper is the formal establishment of an asymptotic separation between these two functions for non-ultimately periodic words, culminating in the main theorem:

$\lim_{n \rightarrow \infty} \frac{\Pal(n) \log (\Pal(n)+1)}{\rho(n)} = 0$

This result not only strengthens previously conjectured bounds but also demonstrates the essential optimality of the numerator.

Preliminaries and Related Work

Factor complexity is known to distinguish between ultimately periodic and aperiodic infinite words: for aperiodic words, $\rho(n)$ is strictly increasing and unbounded [Morse & Hedlund, 1938]. The classical results such as Fekete's lemma ensure the subadditivity of $\log \rho(n)$ and the existence of its asymptotic growth rate. Grillenberger [1973] constructed examples with arbitrarily high complexity, while Sturmian words exhibit minimal, linear complexity.

Prior work relating palindrome complexity to factor complexity provides upper bounds, but these are insufficient to prove the main asymptotic separation in this paper. For instance, Allouche et al. [2003] showed $\Pal(n) < \frac{16}{n} \rho(n+\lfloor n/4 \rfloor)$ for aperiodic words, and Baláži et al. [2007] established various inequalities for uniformly recurrent words, contingent on closure under reversal.

Main Theorem and Proof Strategy

The main theorem asserts that, for any infinite word not ultimately periodic, the ratio $\frac{\Pal(n) \log (\Pal(n)+1)}{\rho(n)}$ approaches zero as $\Pal(n)$0 increases. The proof involves two cases depending on the exponential growth rate $\Pal(n)$1 of $\Pal(n)$2:

• Case $\Pal(n)$3 (Exponential Growth): $\Pal(n)$4 is bounded above by $\Pal(n)$5, making the ratio decay exponentially.
• Case $\Pal(n)$6 (Subexponential Growth): By constructing nearly disjoint sets of $\Pal(n)$7-length factors associated with each palindromic factor and bounding an "exceptional set" of overlaps, it is shown that for arbitrarily large $\Pal(n)$8, $\Pal(n)$9, where $n$0 can be made negligible compared to $n$1. This yields the desired asymptotic.

Technical lemmas, including a result on palindromic prefixes inducing periods, are leveraged to control dependencies among palindromic factors and ordinary factors. The construction shows $n$2 cannot dominate $n$3 asymptotically, even after logarithmic amplification.

Optimality and Construction

The paper rigorously establishes the essential optimality of the numerator. Specifically, it proves that for any nondecreasing function $n$4:

• $n$5 for all non-ultimately periodic infinite words if and only if $n$6.

Moreover, if $n$7 is unbounded, explicit constructions yield infinite words for which the limsup of the ratio diverges. The detailed combinatorial construction exploits periodic blocks separated by large runs of zeros to maximize palindromic occurrences for given factor complexity, overcoming previous conjectures such as whether $n$8 for all infinite words (which is disproved).

Implications and Future Directions

The implications are both combinatorial and algorithmic. Palindromic structures, while prevalent, are strictly subdominant in complexity compared to general factors, even when logarithmically amplified. This impacts algorithms for palindromic factorization, compressed data representations, and symbolic dynamics, showing that "palindrome richness" is asymptotically negligible for non-periodic cases.

The results delimit the boundary for growth rates of palindromic complexity relative to factor complexity, suggesting that further improvements would require fundamentally new invariants or deeper structural conditions. Future research may address complexity in higher-dimensional or non-linear symbolic systems, further analyze the interplay with recurrence properties, or extend to automatic sequences and their combinatorial spectra.

Conclusion

The study provides a sharp asymptotic bound and optimality characterization for the ratio of palindrome complexity to factor complexity in infinite words. It demonstrates that palindromic factors, even when amplified logarithmically, are asymptotically negligible compared to general factor complexity in all non-ultimately periodic words. The theoretical analysis is complemented by explicit combinatorial constructions, setting a reference point for future exploration in the combinatorics of words and symbolic dynamics.