Limiting Behavior of Infinite Products Scaled by Pisot Numbers

Abstract

For $$\theta >1$$ , the infinite product $$\Gamma _{\theta }(x)=\prod _{n=0}^{\infty }\cos (\pi \theta ^{-j}x)$$ is the Fourier transform of the Bernoulli convolution with scale $$\theta ^{-1}$$ . Its limiting behavior at infinity has been studied since the 1930’s, but is still not completely settled. In this note, we consider the limiting behavior of $$\Gamma (x)=\Gamma _{\theta _{1}} (x)\Gamma _{\theta _{2}}(\lambda x)$$ , a question originally raised by Salem. For Pisot numbers $$\theta _{1},\theta _{2}$$ that are exponentially commensurable, we show that the parameters $$\lambda $$ such that $$\Gamma (x)$$ does not tend to zero at infinity are countable, and in most cases they are dense in $${\mathbb {R}}$$ . The explicit forms of such $$\lambda $$ can also be identified. The conclusion is also true for $$\Gamma (x)$$ with n products.