Abstract
We prove that the Thue–Morse sequence t along subsequences indexed by ⌊n c⌋ is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1}L, where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence \(n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}\) has asymptotic density 2−L. This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.
In particular, this result shows that for 1 < c < 3/2 the sequence \(n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}\) attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.
In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue–Morse sequence, that is, it satisfies a Bombieri–Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1}L is contained as an arithmetic subsequence of t.
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Müllner, C., Spiegelhofer, L. Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II. Isr. J. Math. 220, 691–738 (2017). https://doi.org/10.1007/s11856-017-1531-x
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DOI: https://doi.org/10.1007/s11856-017-1531-x