32 research outputs found
Thue--Morse along the sequence of cubes
The Thue--Morse sequence is an automatic sequence over the
alphabet . It can be defined as the binary sum-of-digits function
, reduced modulo , or by using the
substitution , . We prove that the asymptotic density
of the set of natural numbers satisfying equals .
Comparable results, featuring asymptotic equivalence along a polynomial as in
our theorem, were previously only known for the linear case [A. O. Gelfond,
Acta Arith. 13 (1967/68), 259--265], and for the sequence of squares. The main
theorem in [C. Mauduit and J. Rivat, Acta Math. 203 (2009), no. 1, 107--148]
was the first such result for the sequence of squares.
Concerning the sum-of-digits function along polynomials of degree at
least three, previous results were restricted either to lower bounds (such as
for the numbers ), or to sum-of-digits functions in
``sufficiently large bases''. By proving an asymptotic equivalence for the case
of the Thue--Morse sequence, and a cubic polynomial, we move one step closer to
the solution of the third Gelfond problem on the sum-of-digits function
(1967/1968), op. cit.Comment: 50 pages. Corrected several small inconsistencies present in the
first version; reworked the articl