Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer. "Piatetski-Shapiro sequences via Beatty sequences." Acta Arithmetica 166.3 (2014): 201-229. <http://eudml.org/doc/279131>.

@article{LukasSpiegelhofer2014,
abstract = {Integer sequences of the form $⌊n^c⌋$, where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference $∑_\{n≤x\}φ(⌊n^c⌋) - 1/c ∑_\{n≤x^c\} φ(n) n^\{1/c-1\}$, which measures the deviation of the mean value of φ on the subsequence $⌊n^c⌋$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $⌊n^c⌋$ attains both of its values with asymptotic density 1/2.},
author = {Lukas Spiegelhofer},
journal = {Acta Arithmetica},
keywords = {Thue-Morse sequence; Beatty sequences; Piatetski-Shapiro sequences},
language = {eng},
number = {3},
pages = {201-229},
title = {Piatetski-Shapiro sequences via Beatty sequences},
url = {http://eudml.org/doc/279131},
volume = {166},
year = {2014},
}

TY - JOUR
AU - Lukas Spiegelhofer
TI - Piatetski-Shapiro sequences via Beatty sequences
JO - Acta Arithmetica
PY - 2014
VL - 166
IS - 3
SP - 201
EP - 229
AB - Integer sequences of the form $⌊n^c⌋$, where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference $∑_{n≤x}φ(⌊n^c⌋) - 1/c ∑_{n≤x^c} φ(n) n^{1/c-1}$, which measures the deviation of the mean value of φ on the subsequence $⌊n^c⌋$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $⌊n^c⌋$ attains both of its values with asymptotic density 1/2.
LA - eng
KW - Thue-Morse sequence; Beatty sequences; Piatetski-Shapiro sequences
UR - http://eudml.org/doc/279131
ER -