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