Let β ∈ (1, 2) and let Hβ denote Garsia’s entropy for the Bernoulli convolution μβ associated with β. In the present article we show that Hβ > 0.82 for all β ∈ (1, 2) and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields (Formula presented.) for all β ∈ (1, 2). In addition, we show that if an algebraic β is such that (Formula presented.) for some k ⩾ 2, then (Formula presented.). Such is, for instance, any root of a Pisot number which is not a Pisot number itself.