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.