TY - JOUR
T1 - A Lower Bound for the Dimension of Bernoulli Convolutions
AU - Hare, Kevin G.
AU - Sidorov, Nikita
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - Bernoulli convolution
KW - Garsia’s entropy
UR - http://www.scopus.com/inward/record.url?scp=85017135755&partnerID=8YFLogxK
U2 - 10.1080/10586458.2017.1301841
DO - 10.1080/10586458.2017.1301841
M3 - Article
AN - SCOPUS:85017135755
SN - 1058-6458
SP - 1
EP - 5
JO - Experimental Mathematics
JF - Experimental Mathematics
ER -