The local dimensions of the

Bernoulli convolution associated with

the golden number

Author:
Tian-You Hu

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2917-2940

MSC (1991):
Primary 28A80; Secondary 42A85

DOI:
https://doi.org/10.1090/S0002-9947-97-01474-8

MathSciNet review:
1321578

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of i.i.d. random variables each taking values of 1 and with equal probability. For satisfying the equation , let be the probability measure induced by . For any in the range of , let

be the local dimension of at whenever the limit exists. We prove that

where , are respectively the maximum and minimum values of the local dimensions. If , then is the golden number, and the approximate numerical values are and .

**[AZ]**J. C. Alexander and D. Zagier,*The entropy of a certain infinitely convolved Bernoulli measure*, J. London Math. Soc. (2)**44**(1991), 121-134. MR**92g:28035****[AY]**J. C. Alexander and J. A. Yorke,*Fat baker's transformations*, Ergod. Theory & Dynam. Systems**4**(1984), 1-23. MR**86c:58090****[BDGPS]**M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and

J. P. Schreiber,*Pisot and Salem numbers*, Birkhäuser-Verlag, Basel, 1992. MR**93k:11095****[E]**P. Erdös,*On the smoothness properties of a family of Bernoulli convolutions*, Amer. J. Math.**62**(1940), 180-186. MR**1:139e****[F]**K. J. Falconer,*Fractal geometry, mathematical foundations and applications*, Wiley, 1990. MR**92j:28008****[G]**A. M. Garsia,*Arithmetic properties of Bernoulli convolutions*, Trans. Amer. Math. Soc.**102**(1962), 409-432. MR**25:1409****[GH]**J. S. Geronimo and D. P. Hardin,*An exact formula for the measure dimension associated with a class of piecewise linear maps*, Constr. Approx.**5**(1989), 89-98. MR**90d:58076****[JW]**B. Jessen and A. Wintner,*Distribution functions and the Riemann zeta function*, Trans. Amer. Math. Soc.**38**(1935), 48-88.**[HL1]**T. Hu and K. Lau,*The sum of Rademacher functions and Hausdorff dimension*, Math. Proc. Cambridge Philos. Soc.**108**(1990), 97-103. MR**91d:28020****[HL2]**-,*Hausdorff dimension of the level sets of Rademacher series*, Bull. Polish Acad. Sci. Math.**41**(1993), No. 1, 11-18. CMP**96:16****[L1]**K. Lau,*Fractal measure and mean p-variations*, J. Funct. Anal.**108**(1992), No. 2, 427-457. MR**93g:28007****[L2]**-,*Dimension of a family of singular Bernoulli convolutions*, J. Funct. Anal.**116**(1993), 335-358. MR**95h:28013****[LN1]**K. Lau and S. Ngai,*Multifractal measure and a weak separation condition*, Advances in Math., to appear.**[LN2]**-,*The -dimension of the Bernoulli convolution associated with the golden number*, preprint.**[LP]**F. Ledrappier and A. Porzio,*A dimension formula for Bernoulli convolutions*, J. Statist. Phys.**76**(1994), 1307-1327. MR**95i:58111****[PU]**F. Przytycki and M. Urbanski,*On the Hausdorff dimension of some fractal sets*, Studia Math.**93**(1989), 155-186. MR**90f:28006****[Sa]**R. Salem,*Algebraic numbers and Fourier analysis*, Heath, 1963. MR**28:1169****[Si]**C. L. Siegel,*Algebraic integers whose conjugates lie in the unit circle*, Duke Math. J.**11**(1944), 597-602. MR**6:39b****[So]**B. Solomyak,*On the random series (an Erdös problem)*, Ann. of Math.**142**(1995), 611-625. MR**97d:11125****[St]**R. S. Strichartz,*Self-similar measure and their Fourier transformations. III*, Indiana Univ. Math. J.**42**(1993), 367-411. MR**94j:42025**

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Additional Information

**Tian-You Hu**

Affiliation:
Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311

Email:
HUT@gbms01.uwgb.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01474-8

Keywords:
Bernoulli convolution,
Fibonacci sequence,
local dimension,
PV-number

Received by editor(s):
August 23, 1994

Received by editor(s) in revised form:
January 25, 1995

Article copyright:
© Copyright 1997
American Mathematical Society