# The Rudin-Shapiro sequence and similar sequences are normal along squares

## TL;DR

This paper proves that certain digital sequences, including the Rudin-Shapiro sequence, are normal along squares, providing explicit and efficiently generatable normal numbers for any base.

## Contribution

It establishes the normality of digital sequences along squares, encompassing sequences like sum of digits and Rudin-Shapiro, which was previously unknown.

## Key findings

- Sequences like sum of digits modulo m are normal along squares.
- Rudin-Shapiro sequence is normal along squares.
- Normal numbers can be explicitly constructed and efficiently generated.

## 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.