## Lecture Summaries

• Lecture 1, January 13, 1998. Introduction to the course. Using formal languages to solve problems in number theory and vice-versa. Number representations. Algebraicity of the Thue-Morse sequence power series. (See the survey on the subject.)

• Lecture 2, January 15, 1998. Introduction to the course, part II. The Thue-Morse sequence. Automatic sequences. Continued fractions. Fixed points of homomorphisms. The Kolakoski word. k-regular sequences. (See the survey on the subject.)

• Lecture 3, January 20, 1998. The theorems of Lyndon-Schutzenberger, and square-free and overlap-free sequences. (Sections 1.4 and 1.5 of the course notes). Problem Set 1 distributed.

• Lecture 4, January 22, 1998. Introduction to numeration systems. The greedy algorithm. Legendre's theorem on divisibility of n! by primes. Sums of digits. (Sections 3.1 and 3.2 of the course notes).

• Lecture 5, January 27, 1998. Block counting. The Rudin-Shapiro sequence. Representations in negative bases. (Sections 3.3 and 3.6 of the course notes).

• Lecture 6, January 29, 1998. Representation in complex bases. (Section 3.9 of the course notes).

• Lecture 7, February 3, 1998. Introduction to automatic sequences. (Chapter 4 of the course notes.)

• Lecture 8, February 5, 1998. Automatic sequences, iterated homomorphisms, and the k-kernel. (Chapter 5 of the course notes.) Problem Set 2 distributed.

• Lecture 9, February 10, 1998. More on automatic sequences. A sequence is k-automatic iff it is k^n-automatic (Thm. 5.4.2). If u(n) is k-automatic so is u(an+b) (Thm. 7.1.1). If u(n) is k-automatic then so is u(n-1) (Thm. 7.1.2). If u(an+i) is k-automatic for all i, 0 <= i < a, then u(n) is k-automatic. (Thm. 7.1.4).

There was no class on February 12 due to instructor's illness.

There was no class on February 16 because of Winter study period.

• Lecture 10, February 19, 1998. Proof of Lemma 7.1.5 and Theorem 7.1.6 (automatic sequences closed under uniform transductions). Example of an automatic sequence not closed under a (non-uniform) homomorphism. Introduction to formal power series, finite fields, algebraic and transcendental elements.

• Lecture 11, February 24, 1998. Examples of some algebraic formal series: the so-called ``Fredholm'' series, the Cantor series, and the Baum-Sweet series. Example of a transcendental formal series. Lemmas for the proof of the Christol theorem (Lemmas 12.2.1, 12.2.2 of the text). Problem Set 3 distributed.

• Lecture 12, February 26, 1998. The proof of the Christol theorem, showing formal series algebraic over GF(q)[X] correspond to q-automatic sequences (Theorem 12.2.3). Proof that pi_q is transcendental (Sections 12.4 and 12.5).

• Lecture 13, March 3, 1998. Applications of the Christol theorem. Proof that the theta function is transcendental over Q(x). Proof that the primitive words are not unambiguously context-free. Proof of closure under Hadamard product. Introduction to the Carlitz-Goss p-adic gamma function.

• Lecture 14, March 5, 1998. Proof of the transcendence of the Carlitz-Goss p-adic gamma function, following Mendès France and Yao (J. Number Theory 63 (1997), 396-402). Introduction to fixed points of homomorphisms.

• Lecture 15, March 10, 1998. Finished up the proof of the characterization of fixed points of homomorphisms.

• Lecture 16, March 12, 1998. Images of fixed points of homomorphisms can be made epsilon-free; closure under shift. Introduction to Sturmian words (Chapter 9).

• Lecture 17, March 17, 1998. More on Sturmian words (sections 9.1-9.3 of the course notes).

• Lecture 18, March 19, 1998. Subword complexity. Sections 10.1-10.2, 10.5 of the course notes.

• Lecture 19, March 24, 1998. Finished the proof that Sturmian words have subword complexity n+1. Statement and proof of one direction of Mignosi's theorem. Introduction to automatic real numbers (Chapter 13 of the course notes). Lehr's proof that L(k,b) forms a vector space over the rationals.

• Lecture 20, March 26, 1998. Transcendence theory. Proof of transcendence of sum B^(-2^k). Liouville inequality. Mahler's method.

• Lecture 21, March 31, 1998. More on transcendence. Transcendence of characteristic numbers using Roth's theorem. Transcendence of binary numbers that are fixed points of primitive morphisms or morphisms of constant length, part I.

• Lecture 22, April 2, 1998. Finishing the proof of the Allouche-Zamboni result on transcendence of binary homomorphic numbers; Seebold's result.

shallit@graceland.uwaterloo.ca