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Transactions of the American Mathematical Society

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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
MathSciNet review: 1321578
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $X_1,X_2,\dotsc $ be a sequence of i.i.d. random variables each taking values of 1 and $-1$ with equal probability. For $1/2<\rho <1$ satisfying the equation $1-\rho -\dotsb -\rho ^s=0$, let $\mu $ be the probability measure induced by $S=\sum _{i=1}^\infty \rho ^iX_i$. For any $x$ in the range of $S$, let

\begin{displaymath}d(\mu ,x)=\lim _{r\to 0^+}\log \mu([x-r,x+r])/\log r\end{displaymath}

be the local dimension of $\mu $ at $x$ whenever the limit exists. We prove that

\begin{displaymath}\alpha ^*=-\frac {\log 2}{\log \rho}\quad \text{and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho}-\frac {\log 2}{\log \rho},\end{displaymath}

where $\delta =(\sqrt {5}-1)/2$, are respectively the maximum and minimum values of the local dimensions. If $s=2$, then $\rho $ is the golden number, and the approximate numerical values are $\alpha ^*\approx 1.4404$ and $\alpha _*\approx 0.9404$.

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  • [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 $L^q$-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 $\Sigma \pm\lambda ^n$ (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

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

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