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# The Rudin-Shapiro Sequence and Similar Sequences are Normal Along Squares

Published:2018-04-30
Printed: Oct 2018
• Clemens Müllner,
Institut für Diskrete Mathematik und Geometrie TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
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## Abstract

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.
 Keywords: Rudin-Shapiro sequence, digital sequence, normality, exponential sum
 MSC Classifications: 11A63 - Radix representation; digital problems {For metric results, see 11K16} 11B85 - Automata sequences 11L03 - Trigonometric and exponential sums, general 11N60 - Distribution functions associated with additive and positive multiplicative functions 60F05 - Central limit and other weak theorems

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