Spectral structure of digit sets of self-similar tiles on ${Bbb R}^1$

TL;DR

This paper analyzes the spectral structure of digit sets for self-similar tiles in one dimension, revealing new algebraic characterizations using cyclotomic polynomials beyond known classes.

Contribution

It introduces a spectral approach to characterize tile digit sets via the zeros of their mask polynomials, extending the class of known digit sets with a cyclotomic polynomial framework.

Findings

Characterization of tile digit sets through the spectrum of their mask polynomials.

Extension of the product-form class to higher order using cyclotomic polynomials.

Connection between spectral properties and algebraic structure of digit sets.

Abstract

We study the structure of the digit sets ${\mathcal D}$ for the integral self-similar tiles $T(b,{\mathcal{D}})$ (we call such ${\mathcal D}$ a {\it tile digit set} with respect to $b$). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial $P_{\mathcal D}$, i.e., the zeros of $P_{\mathcal D}$ on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomial, we characterize the tile digit sets through some product of cyclotomic polynomials (kernel polynomials), which is a generalization of the product-form to higher order.