Kevin G. Hare

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos. Joo and Komornik in 1990. is the determination of l(q) for Pisot numbers q, where l(q) = inf(|y|:y = e0 + e1q1 + ...+ enqn, ei ∈ {±1,0}, y ≠ 0.) Although the quantity l(q) is known for some Pisot numbers q, there has been no general method for computing l(q). This paper gives such an algorithm. With this algorithm, some properties of l(q) and its generalizations are investigated.A related question concerns the analogy of l(q), denoted a(q), where the coefficients are restricted to ±1; in particular, for which non-Pisot numbers is a(q) nonzero? This paper finds an infinite class of Salem numbers where a(q) ≠ 0.

On univoque Pisot numbers

We study Pisot numbers

New techniques for bounds on the total number of prime factors of an odd perfect number

\beta \in (1, 2)

The minimal growth of a \(k\)-regular sequence

which are univoque, i.e., such that there exists only one representation of

More on the total number of prime factors of an odd perfect number

1

Stolarsky’s conjecture and the sum of digits of polynomial values

as

BETA-EXPANSIONS OF PISOT AND SALEM NUMBERS

1 = \sum_{n \geq 1} s_n\beta^{-n}

Negative Pisot and Salem numbers as roots of Newman polynomials

, with

A lower bound for Garsia's entropy for certain Bernoulli convolutions

s_n \in \{0, 1\}

General Forms for Minimal Spectral Values For a Class of Quadratic Pisot Numbers

. We prove in particular that there exists a smallest univoque Pisot number, which has degree