Refinable Functions with PV Dilations

Abstract

A PV number is an algebraic integer α of degree d ≥ 2 all of whose Galois conjugates other than itself have modulus less than 1. Erdös [8] proved that the Fourier transform ϕ, of a nonzero compactly supported scalar valued function satisfying the refinement equation ϕ(x) = |α| 2 ϕ(αx) + |α| 2 ϕ(αx − 1) with P V dilation α, does not vanish at infinity so by the Riemann-Lebesgue lemma ϕ is not integrable. Dai, Feng and Wang [5] extended his result to scalar valued solutions of ϕ(x) = k a(k)ϕ(αx − τ (k)) where τ (k) are integers and a has finite support and sums to |α|. In ([22], Conjecture 4.2) we conjectured that their result holds under the weaker assumption that τ has values in the ring of polynomials in α with integer coefficients. This paper formulates a stronger conjecture and provides support for it based on a solenoidal representation of ϕ, and deep results of Erdös and Mahler [9];Odoni [26] that give lower bounds for the asymptotic density of integers represented by integral binary forms of degree > 2;degree = 2, respectively. We also construct an integrable vector valued refinable function with PV dilation.