The nonintegrability conjecture for PV refinable distributions

Let $\lambda$ be a Pisot–Vijayaraghavan (PV) number, let $f$ be a nonzero refinable distribution for dilation by $\lambda$, and suppose its translations lie in

$$\mathbb{Z}[1,\lambda,\ldots,\lambda^{n-1}].$$

Here $\mathbb{Q}[\lambda]$ denotes the field generated by $\lambda$ over $\mathbb{Q}$, and $\widehat f$ is the Fourier transform of $f$. PV nonintegrability conjecture. There exists $\alpha\in\mathbb{Q}[\lambda]\setminus\{0\}$ and $\gamma\in\mathbb{C}\setminus\{0\}$ such that

$$\lim_{k\rightarrow\infty}\widehat f(\lambda^k\alpha)=\gamma.$$

Consequently, the Riemann–Lebesgue lemma implies that $f$ is not integrable. This conjecture is proposed as an extension of results of Erdős, Kahane, and Dai–Feng–Wang to certain noninteger translation values.