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Piatetski-Shapiro sequences via Beatty sequences

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Spiegelhofer L.

Acta Arithmetica

2014

tom 166

nr 3

201-229

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.

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1 Acta Arithmetica

1 Spiegelhofer L.

1 2014

Wyniki wyszukiwania

Piatetski-Shapiro sequences via Beatty sequences