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L. Spiegelhofer, C. Müllner:
"Normality of the Thu-Morse sequence along Piatetski-Shapiro sequences";
Quarterly Journal of Mathematics, 66 (2015), 1127 - 1138.

We prove that the Thue--Morse sequence t along subsequences indexed by ⌊nc⌋ 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 subword (contiguous finite subsequence) of the sequence n↦t⌊nc⌋ 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↦t⌊nc⌋ 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.

http://dx.doi.org/10.1093/qmath/hav029