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 February 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 February 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).
Lecture 3 on March 7, 2025. I stated Lemma 6A, Theorem 7A, Theorem 8A, Lemma 9A and Theorem 10A. Then I went,
partially but I will not return to this, through the proof of Roth's inequality (Roth's first theorem): If a is
a real irrational algebraic integer and ep > 0, then the inequality |a - p/q| < q^{-2 - ep} has only
finitely many rational solutions p/q, q > 0.
Lecture 4 on March 14, 2025. I stated the Theorem (Szemeredi, 1975): Let m > 2 be an integer. Then for
every de > 0 there is an integer N > 0 such that if n > N and X is a subset of [n]
(= {1, 2, ..., n}) with |X| > de*n, then X contains an m-term arithmetic progression.
The second theorem of Roth (1952) is Szemeredi's theorem for m = 3. I started to present Szemeredi's combinatorial
proof of it. Proposition 1: Every de-sequence contains a 3-term AP; this is just an equivalent restatement of
the previous theorem. Lemma 2: Let l > 0 be an integer. Then for every de > 0 every de-sequence
contains an l-cube; proof. Lemma 3: Roughly stated, every de-sequence has a saturated
de'-subsequence, where de' is at least de; proof. I introduced d-decompositions of finite sets of
natural numbers and will continue with Lemma 4 on them next time.
Lecture 5 on March 21, 2025. Lemma 4 on d-decompositions. The combinatorial proof (of E. Szemeredi) of Roth's
theorem on 3-term APs (here formulated so that every de-sequence contains a 3-term AP).
Lecture 6 on March 28, 2025. The first part of the analytic proof of Roth's
theorem on 3-term APs. I follow my older lecture notes from 2010.
Lecture 7 on April 4, 2025. The second part of the analytic proof of Roth's
theorem on 3-term APs.
April 2025