Refinable functions with PV dilations

A PV number is an algebraic integer $\alpha$ of degree $d \geq 2$ all of whose Galois conjugates other than itself have modulus less than $1$. Erd\"{o}s \cite{erdos} proved that the Fourier transform $\widehat \varphi,$ of a nonzero compactly supported scalar valued function satisfying the refinement equation $\varphi(x) = \frac{|\alpha|}{2}\varphi(\alpha x) + \frac{|\alpha|}{2}\varphi(\alpha x-1)$ with $PV$ dilation $\alpha,$ does not vanish at infinity so by the Riemann-Lebesgue lemma $\varphi$ is not integrable. Dai, Feng and Wang \cite{daifengwang} extended his result to scalar valued solutions of $\varphi(x) = \sum_k a(k) \varphi(\alpha x - \tau(k))$ where $\tau(k)$ are integers and $a$ has finite support and sums to $|\alpha|$. In (\cite{lawton3}, Conjecture 4.2) we conjectured that their result holds under the weaker assumption that $\tau$ has values in the ring of polynomials in $\alpha$ with integer coefficients. This paper formulates a stronger conjecture and provides support for it based on a solenoidal representation of $\widehat \varphi,$ and deep results of Erd\"{o}s 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.