## The local dimensions of the Bernoulli convolution associated with the golden number

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- by Tian-You Hu PDF
- Trans. Amer. Math. Soc.
**349**(1997), 2917-2940 Request permission

## 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 \[ d(\mu ,x)=\lim _{r\to 0^+}\log \mu ([x-r,x+r])/\log r\] be the local dimension of $\mu$ at $x$ whenever the limit exists. We prove that \[ \alpha ^*=-\frac {\log 2}{\log \rho }\quad \text {and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho }-\frac {\log 2}{\log \rho },\] 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 $\alpha _*\approx 0.9404$.## References

- J. C. Alexander and Don Zagier,
*The entropy of a certain infinitely convolved Bernoulli measure*, J. London Math. Soc. (2)**44**(1991), no. 1, 121–134. MR**1122974**, DOI 10.1112/jlms/s2-44.1.121 - J. C. Alexander and J. A. Yorke,
*Fat baker’s transformations*, Ergodic Theory Dynam. Systems**4**(1984), no. 1, 1–23. MR**758890**, DOI 10.1017/S0143385700002236 - M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber,
*Pisot and Salem numbers*, Birkhäuser Verlag, Basel, 1992. With a preface by David W. Boyd. MR**1187044**, DOI 10.1007/978-3-0348-8632-1 - T. Venkatarayudu,
*The $7$-$15$ problem*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 531. MR**0000001**, DOI 10.1090/gsm/058 - Kenneth Falconer,
*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677** - Adriano M. Garsia,
*Arithmetic properties of Bernoulli convolutions*, Trans. Amer. Math. Soc.**102**(1962), 409–432. MR**137961**, DOI 10.1090/S0002-9947-1962-0137961-5 - J. S. Geronimo and D. P. Hardin,
*An exact formula for the measure dimensions associated with a class of piecewise linear maps*, Constr. Approx.**5**(1989), no. 1, 89–98. Fractal approximation. MR**982726**, DOI 10.1007/BF01889600 - B. Jessen and A. Wintner,
*Distribution functions and the Riemann zeta function*, Trans. Amer. Math. Soc.**38**(1935), 48–88. - Tian You Hu and Ka-Sing Lau,
*The sum of Rademacher functions and Hausdorff dimension*, Math. Proc. Cambridge Philos. Soc.**108**(1990), no. 1, 97–103. MR**1049763**, DOI 10.1017/S0305004100068985 - —,
*Hausdorff dimension of the level sets of Rademacher series*, Bull. Polish Acad. Sci. Math.**41**(1993), No. 1, 11–18. - Ka-Sing Lau,
*Fractal measures and mean $p$-variations*, J. Funct. Anal.**108**(1992), no. 2, 427–457. MR**1176682**, DOI 10.1016/0022-1236(92)90031-D - Ka-Sing Lau,
*Dimension of a family of singular Bernoulli convolutions*, J. Funct. Anal.**116**(1993), no. 2, 335–358. MR**1239075**, DOI 10.1006/jfan.1993.1116 - K. Lau and S. Ngai,
*Multifractal measure and a weak separation condition*, Advances in Math., to appear. - —,
*The $L^q$-dimension of the Bernoulli convolution associated with the golden number*, preprint. - François Ledrappier and Anna Porzio,
*A dimension formula for Bernoulli convolutions*, J. Statist. Phys.**76**(1994), no. 5-6, 1307–1327. MR**1298104**, DOI 10.1007/BF02187064 - F. Przytycki and M. Urbański,
*On the Hausdorff dimension of some fractal sets*, Studia Math.**93**(1989), no. 2, 155–186. MR**1002918**, DOI 10.4064/sm-93-2-155-186 - Raphaël Salem,
*Algebraic numbers and Fourier analysis*, D. C. Heath and Company, Boston, Mass., 1963. MR**0157941** - Albert Eagle,
*Series for all the roots of the equation $(z-a)^m=k(z-b)^n$*, Amer. Math. Monthly**46**(1939), 425–428. MR**6**, DOI 10.2307/2303037 - Boris Solomyak,
*On the random series $\sum \pm \lambda ^n$ (an Erdős problem)*, Ann. of Math. (2)**142**(1995), no. 3, 611–625. MR**1356783**, DOI 10.2307/2118556 - Robert S. Strichartz,
*Self-similar measures and their Fourier transforms. III*, Indiana Univ. Math. J.**42**(1993), no. 2, 367–411. MR**1237052**, DOI 10.1512/iumj.1993.42.42018

## Additional Information

**Tian-You Hu**- Affiliation: Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311
- Email: HUT@gbms01.uwgb.edu
- Received by editor(s): August 23, 1994
- Received by editor(s) in revised form: January 25, 1995
- © Copyright 1997 American Mathematical Society
- 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