Comments on the height reducing property

Abstract

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F , of the ring Z of the rational integers such that. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.

Comments on the height reducing property II

“…Letting L be the absolute norm of the denominator of the fractional ideal (α ℓ ) in Q(α), we obtain LJ (n) (β) ∈ Lα ℓ Z[1/α] ∩ Z[α], and we can deduce the result similarly to the proof of [7,Lemma 3].…”

“…. , ε n ) ∈ S n+1 , then we see, by (7), that β = R(α) for some R(x) := . Now, consider α ∈ F 2 which is not a root of unity.…”

“…Notice that Theorem 2 (ii) can also be applied in the non-integral case. For example, if M α (x) = 2x 2 − 3x + 2, then α is a quadratic algebraic number whose conjugates are of modulus 1 (α satisfies HRP by [7,Theorem 2]) and α / ∈ F 2 , since M α (1) = 1.…”

“…Recall that a complex number α is said to satisfy HRP if there is a finite set F ⊂ Z, such that each polynomial with coefficients in Z, evaluated at α, belongs to the set (see e.g. [3,6,7]). In other words, α satisfies HRP when Z[α] may be reduced to F [α].…”

Minimal Non-Integer Alphabets Allowing Parallel Addition

“…): By Lemma 2.2, β is an algebraic number since A[β] ⊂ Fin A (β). The proof that β must be expanding is based on the paper of S. Akiyama and T. Zäimi [9]. Let β be an algebraic conjugate of β and σ : Q(β) → Q(β ) be the field isomorphism such that σ(β) = β .…”