# Automatic Sequences - Theory, Applications, Generalizations

@inproceedings{Allouche2003AutomaticS, title={Automatic Sequences - Theory, Applications, Generalizations}, author={Jean-Paul Allouche and Jeffrey Shallit}, year={2003} }

Preface 1. Stringology 2. Number theory and algebra 3. Numeration systems 4. Finite automata and other models of computation 5. Automatic sequences 6. Uniform morphisms and automatic sequences 7. Morphic sequences 8. Frequency of letters 9. Characteristic words 10. Subwords 11. Cobham's theorem 12. Formal power series 13. Automatic real numbers 14. Multidimensional automatic sequences 15. Automaticity 16. k-regular sequences 17. Physics Appendix. Hints, references and solutions for selected…

## 489 Citations

### Finite Automata and Algebraicity of Formal Power Series over Finite Fields

- Mathematics, Computer Science
- 2017

This project is intended to study the relations of automatic sequences with algebra. To this end, we first look at stringology. We study finite-state automata and then automatic sequences that are…

### Automatic sequences as good weights for ergodic theorems

- Mathematics, Computer Science
- 2017

It is shown that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems, and for totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodics theorem in L^1 holds.

### Distribution Modulo One and Diophantine Approximation

- Mathematics
- 2012

1. Distribution modulo one 2. On the fractional parts of powers of real numbers 3. On the fractional parts of powers of algebraic numbers 4. Normal numbers 5. Further explicit constructions of normal…

### Univoque Numbers and Automatic Sequences

- Mathematics
- 2010

A set of binary sequences related to the iteration of unimodal continuous functions of the interval [0,1] appears in a 1982–1983 work of Cosnard and the first author. An almost identical set of…

### Number Theoretic Aspects of Regular Sequences

- Mathematics
- 2018

We present a survey of results concerning regular sequences and related objects. Regular sequences were defined in the early 1990s by Allouche and Shallit as a combinatorially, algebraically, and…

### More on Generalized Automatic Sequences

- Computer Science, MathematicsJ. Autom. Lang. Comb.
- 2002

Some generalizations of k-automatic sequences replacing the k-ary system by an abstract numeration system on a regular language are given and the equivalence of these sequences and morphic predicates is given.

### Mix-Automatic Sequences

- Computer Science, MathematicsLATA
- 2013

This paper adapts the notion of k-kernels to obtain a characterization of mix- automatic sequences, and employs this notion to construct morphic sequences that are not mix-automatic.

### Irrationality measures for some automatic real numbers

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2009

Abstract This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive…

### Outstanding Challenges in Combinatorics on Words ( 12 w 5068 )

- Mathematics
- 2012

Combinatorics on words is a relatively new area of research in discrete mathematics. It studies the properties of words (sequences), either finite or infinite, over a finite alphabet. The perspective…

### Everywhere alpha -Repetitive Sequences and Sturmian Words

- MathematicsCSR
- 2007

This work studies everywhere α-repetitive sequences, sequences in which every position has an occurrence of a repetition of order α ≥ 1 of bounded length, and shows that Sturmian words are among the optimal 2 - and 2+-rePETitive sequences.

## References

SHOWING 1-2 OF 2 REFERENCES

### Page 425: Open Problem 14.8.2 was already solved by A. Carpi, Multidimensional unrepetitive configurations

- Also, the problem should impose the restriction gcd(c, e) = 1.) (N. Rampersad
- 1988