Piatetski-Shapiro sequences via Beatty sequences

• Autorzy: Lukas Spiegelhofer
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11A63: Radix representation; digital problems
• 11B83: Special sequences and polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 166
• Numer: 3
• Strony: 201-229

Daty

wydano

2014

Twórcy

autor

Lukas Spiegelhofer

• Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Identyfikatory

DOI

10.4064/aa166-3-1