Discrete spectra and Pisot numbers

TL;DR

This paper investigates the properties of the m-spectrum of real numbers greater than one, establishing a key connection between accumulation points, the parameter q, and Pisot numbers, with implications for spectral and expansion theories.

Contribution

It proves that the m-spectrum has an accumulation point if and only if q<m+1 and q is not a Pisot number, refining understanding of spectral distributions.

Findings

Y^m(q) has an accumulation point iff q<m+1 and q is not a Pisot number.

Improved results on the distribution of points in the m-spectrum.

Established a precise criterion linking spectral accumulation to Pisot numbers.

Abstract

By the m-spectrum of a real number q>1 we mean the set Y^m(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their intimate connections with infinite Bernoulli convolutions, spectral properties of substitutive point sets and expansions in noninteger bases. We prove that Y^m(q) has an accumulation point if and only if q<m+1 and q is not a Pisot number. Consequently a number of related results on the distribution of points of this form are improved.