1,959 research outputs found
Automatic sequences as good weights for ergodic theorems
We study correlation estimates of automatic sequences (that is, sequences
computable by finite automata) with polynomial phases. As a consequence, we
provide a new class of good weights for classical and polynomial ergodic
theorems, not coming themselves from dynamical systems.
We show that automatic sequences are good weights in for polynomial
averages and totally ergodic systems. For totally balanced automatic sequences
(i.e., sequences converging to zero in mean along arithmetic progressions) the
pointwise weighted ergodic theorem in holds. Moreover, invertible
automatic sequences are good weights for the pointwise polynomial ergodic
theorem in , .Comment: 31 page
Finite automata and algebraic extensions of function fields
We give an automata-theoretic description of the algebraic closure of the
rational function field F_q(t) over a finite field, generalizing a result of
Christol. The description takes place within the Hahn-Mal'cev-Neumann field of
"generalized power series" over F_q. Our approach includes a characterization
of well-ordered sets of rational numbers whose base p expansions are generated
by a finite automaton, as well as some techniques for computing in the
algebraic closure; these include an adaptation to positive characteristic of
Newton's algorithm for finding local expansions of plane curves. We also
conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version,
includes minor edit
On a conjecture of Dekking : The sum of digits of even numbers
Let and denote by the sum-of-digits function in base . For
consider # \{0 \le n < N : \;\;s_q(2n) \equiv j \pmod q \}.
In 1983, F. M. Dekking conjectured that this quantity is greater than
and, respectively, less than for infinitely many , thereby claiming an
absence of a drift (or Newman) phenomenon. In this paper we prove his
conjecture.Comment: 6 pages, accepted by JTN
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