M\"{o}bius orthogonality of Thue-Morse sequence along Piatetski-Shapiro numbers

TL;DR

This paper proves that the M"obius function is orthogonal to the Thue-Morse sequence evaluated along Piatetski-Shapiro numbers for any exponent c between 1 and 2, extending previous results from squares.

Contribution

It extends M"obius orthogonality of Thue-Morse sequence from squares to Piatetski-Shapiro numbers for all c in (1,2).

Findings

M"obius function is orthogonal to Thue-Morse sequence along Piatetski-Shapiro numbers.

Orthogonality holds for all c in (1,2).

Results include sequences with maximum entropy.

Abstract

We show that the M\"{o}bius function is orthogonal to the Thue-Morse sequence $t(n)$ taken along the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for any $1 < c < 2$. Previously this property was established for the subsequence along the squares $t(n^2)$. These are both examples of M\"{o}bius orthogonal sequences with maximum entropy.