For $$\theta >1$$ θ > 1 , the infinite product $$\Gamma _{\theta }(x)=\prod _{n=0}^{\infty }\cos (\pi \theta ^{-j}x)$$ Γ θ ( x ) = ∏ n = 0 ∞ cos ( π θ - j x ) is the Fourier transform of the Bernoulli convolution with scale $$\theta ^{-1}$$ θ - 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 $$\Gamma (x)=\Gamma _{\theta _{1}} (x)\Gamma _{\theta _{2}}(\lambda x)$$ Γ ( x ) = Γ θ 1 ( x ) Γ θ 2 ( λ x ) , a question originally raised by Salem. For Pisot numbers $$\theta _{1},\theta _{2}$$ θ 1 , θ 2 that are exponentially commensurable, we show that the parameters $$\lambda $$ λ such that $$\Gamma (x)$$ Γ ( x ) does not tend to zero at infinity are countable, and in most cases they are dense in $${\mathbb {R}}$$ R . The explicit forms of such $$\lambda $$ λ can also be identified. The conclusion is also true for $$\Gamma (x)$$ Γ ( x ) with n products.