Refinable functions with PV dilations

Abstract

A PV number is an algebraic integer $α$ of degree $d \geq 2$ all of whose Galois conjugates other than itself have modulus less than $1$. Erdös \cite{erdos} proved that the Fourier transform $\widehat φ,$ of a nonzero compactly supported scalar valued function satisfying the refinement equation $φ(x) = \frac{|α|}{2}φ(αx) + \frac{|α|}{2}φ(αx-1)$ with $PV$ dilation $α,$ does not vanish at infinity so by the Riemann-Lebesgue lemma $φ$ is not integrable. Dai, Feng and Wang \cite{daifengwang} extended his result to scalar valued solutions of $φ(x) = \sum_k a(k) φ(αx - τ(k))$ where $τ(k)$ are integers and $a$ has finite support and sums to $|α|$. In (\cite{lawton3}, 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 $\widehat φ,$ and deep results of Erdös and Mahler \cite{erdosmahler};Odoni \cite{odoni} 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.