2015/11/30 by Clemens Müllner, Lukas Spiegelhofer · 1 citation
Computer Science · Mathematics · #Analytic Number Theory Research #Combinatorics #Discrete mathematics #Distribution (mathematics) #Exponent #Longest increasing subsequence #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Morse code #Multiplicative function #Omega #Physics #Sequence (biology) #Subsequence #Type (biology) #Upper and lower bounds #math.NT #msc:11A63 #msc:11B25 #msc:11B83 #msc:11K16 #semigroups and automata theory
paper · pdf · doi:10.1007/s11856-017-1531-x
published as Israel J. Math. 220 (2017), no. 2, 691-738 · 33 pages
openalex created_date 2016/06/24 · openalex publication_date 2017/05/18 · arxiv created 2017/11/15 · arxiv updated 2017/11/16 · openalex updated_date 2026/08/05
We prove that the Thue--Morse sequence \mathbf t along subsequences indexed by \lfloor nc\rfloor 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↦ \mathbf t\lfloor nc\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↦ \mathbf t\lfloor nc\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 \mathbf t.