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arXiv · 1707.05094

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer
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Abstract

Integer sequences of the form $\lfloor n^c\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\lfloor nα+β\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[\sum_{n\leq x}φ\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}φ(n)n^{\frac 1c-1},\] which measures the deviation of the mean value of $φ$ 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 $1/2$.

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Lukas Spiegelhofer. 2017-07-17. Piatetski-Shapiro sequences via Beatty sequences. https://arxiv.org/abs/1707.05094

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