A Lower Bound for the Dimension of Bernoulli Convolutions
A Lower Bound for the Dimension of Bernoulli Convolutions
复制标题
伯努利卷积维数的下界
作者：
K. Hare;N. Sidorov
设β ∈（1，2），Hβ表示与β相关的Bernoulli卷积μβ的Garsia熵.本文证明了对所有β ∈（1，2），Hβ > 0.82，并在一定范围内改进了这一界.结合Hochman和Breuillard-Varjú最近的结果，这产生了所有β ∈（1，2）。此外，我们证明了，如果一个代数β是这样的，对于某些k <$2，那么。这是，例如，任何根的皮索数，这不是皮索数本身。
ABSTRACT 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 for all β ∈ (1, 2). In addition, we show that if an algebraic β is such that for some k ⩾ 2, then . Such is, for instance, any root of a Pisot number which is not a Pisot number itself.