## Seminario

Hillel Furstenberg (Hebrew University of Jerusalem)

Aula 1E, Palazzina RM 004

An equidistribution problem, algebraic functions over finite fields,

and Apery-like sequences

Abstract: A known open problem is the distribution mod 1 of powers

of a rational number p/q with p>q. This can be seen as a question

regarding the behavior of orbits of "special" points in a cellular

automaton. An example of a cellular automaton where this kind of behavior

can be determined is where the space consists of sequences from a finite

field and the cellular operation is a fixed linear combination of

translates The entries at a fixed position along the orbit turn out to be

the coefficient sequence of algebraic functions. A generalization of this

construction leads to the notion of a "dirational" sequence: the sequence of

diagonal terms in the expansion of a rational function of several

variables. These functions are related to modular and hypergeometric

functions, and we show that the sequences playing a major role in Apery's

proof of the irrationality of \zeta(3) are dirational sequences, explaining

thereby why the terms are integers or close to integers.