Limiting Behavior of Infinite Products Scaled by Pisot Numbers

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