Book Review: Recurrence in ergodic theory and combinatorial number theory

Ergodic Theory and Number Theory

Elon Lindenstrauss was awarded the 2010 Fields Medal for his results on measure rigidity in ergodic theory, and their applications to number theory. The web page of the ICM 2010 contains the following brief description of Elon Lindenstrauss’ achievements: Lindenstrauss has made far-reaching advances in ergodic theory, the study of measure preserving transformations. His work on a conjecture of ...

Ergodic Theory: Recurrence

Almost every, essentially: Given a Lebesgue measure space (X,B, μ), a property P (x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x : P (x) holds} has zero measure. Two sets A,B ∈ B are essentially disjoint if μ(A ∩B) = 0. Conservative system: Is an infinite measure preserving system such that for no set A ∈ B with positive measure are A,T−1A,T−2A, . . . p...

Combinatorial and Diophantine Applications of Ergodic Theory

Ergodic Theory and Geometric Rigidity and Number Theory

Scientific background and objectives The central scientific theme of this programme was the recent development of applications of ergodic theory to other areas of mathematics, in particular, the connections with geometry, group actions and rigidity, and number theory. The potential of ergodic theory as a tool in number theory was emphasized by Furstenberg’s proof of Szemerdi’s theorem on arithm...

Combinatorial Number Theory

Background. The integer distance graph G(Z, D) with distance set D = {d1, d2, . . .} has the set of integers Z as the vertex set and two vertices x, y ∈ Z are adjacent if and only if |x − y| ∈ D. The integer distance graphs (under Euclidean norm) were first systematically studied by Eggleton–Erdős–Skilton in 1985 [12, 13], and have been investigated in many ways [50, 56, 57, 61]. One of main go...