Analytic and Combinatorial Number Theory, NDMI045, summer term 2024/25
My plan is to lecture on the following theorems. 1. Roth's theorem on Diophantine approximation.
2. Roth's theorem on three-term APs in dense sets of integers. 3. The PNT (prime number theorem).
4. Dirichlet's theorem on prime numbers in APs. Here AP = arithmetical progression.
lecture notes (preliminary, updated March 6)
Lecture 1 on Feb 21, 2025. Liouville'e inequality (proof), generates transcendental numbers. Thue
equations have only finitely many solutions, in contrast with generalized Pell equations. A reduction: any
nontrivial improvement of Liouville'e inequality yields finiteness of solution sets of Thue equations (an outline
of the proof). Thue's inequality. Roth's inequality (just statement). I will present the proof of Roth's inequality
as it is given in the book W. Schmidt, Diophantine Approximation, Springer-Verlag, Berlin 1980.
Lecture 2 on Feb 28, 2025. I stated Lemmas 4A, 4B, 4C, 5A, 5B and 5C, with the related notions, and I proved
Lemma 5B which is Siegel's lemma. I also stated the reduction of Roth's inequality from algebraic numbers to algebraic integers,
and I defined the index of an integral polynomial P(X_1, ..., X_m) with respect to the 2m-tuple (r_1, ..., r_m;
a_1, ..., a_m).
March 2025