Spectrum of a Rudin-Shapiro-like sequence

TL;DR

This paper demonstrates that a new Rudin-Shapiro-like sequence exhibits a purely singular continuous diffraction spectrum, contrasting with the absolutely continuous spectrum of the classical Rudin-Shapiro sequence, thereby answering an open question.

Contribution

It establishes the spectral type of a recently proposed Rudin-Shapiro-like sequence, showing it has a singular continuous diffraction spectrum.

Findings

The sequence has purely singular continuous diffraction spectrum.

This contrasts with the classical Rudin-Shapiro sequence's absolutely continuous spectrum.

The result answers an open question about the spectral nature of the new sequence.

Abstract

We show that a recently proposed Rudin-Shapiro-like sequence, with balanced weights, has purely singular continuous diffraction spectrum, in contrast to the well-known Rudin-Shapiro sequence whose diffraction is absolutely continuous. This answers a question that had been raised about this new sequence.