Normality of the Thu-Morse sequence along Piatetski-Shapiro sequences

Abstract

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.