Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer¹

¹ Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstrasse 8–10 1040 Wien, Austria

Acta Arithmetica, Tome 166 (2014) no. 3, pp. 201-229

Résumé

Integer sequences of the form $\lfloor n^c\rfloor$, where $1 c 2$, can be locally approximated by sequences of the form $\lfloor n\alpha+\beta\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[ \sum_{n\leq x}\varphi(\lfloor n^c\rfloor) - \frac 1c\sum_{n\leq x^c}\varphi(n)n^{1/c-1}, \] which measures the deviation of the mean value of $\varphi$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1 c\leq 1.42$ the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with asymptotic density $1/2$.

Détail

DOI : 10.4064/aa166-3-1

Keywords: integer sequences form lfloor rfloor where locally approximated sequences form lfloor alpha beta rfloor following approach led estimate difference sum leq varphi lfloor rfloor frac sum leq varphi which measures deviation mean value varphi subsequence lfloor rfloor expected value expression involving exponential sums application prove leq subsequence thue morse sequence indexed lfloor rfloor attains its values asymptotic density