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On the dimension of Bernoulli convolutions for all transcendental parameters. (English) Zbl 1426.28024

Let \(\{X_i\}_{i=0}^\infty\) be a sequence of independent random variables taking the values \(\pm 1\) with equal probability. For a parameter \(\lambda\in (0,1)\), the Bernoulli convolution \(\nu_\lambda\) is the probability measure on \(\mathbb{R}\) which is the law of the random variable \(\sum\limits_{i=0}^\infty \lambda^i X_i\). The author proves that if \(\lambda\) is transcendental in \((0,1)\) then the dimension of the measure \(\nu_\lambda\) equals one.

MSC:

28A80 Fractals
42A85 Convolution, factorization for one variable harmonic analysis
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[1] Akiyama, Shigeki; Feng, De-Jun; Kempton, Tom; Persson, Tomas, On the {H}ausdorff Dimension of {B}ernoulli Convolutions, (2018) · Zbl 1483.28004
[2] Bombieri, Enrico; Gubler, Walter, Heights in {D}iophantine Geometry, New Math. Monogr., 4, xvi+652 pp., (2006) · Zbl 1115.11034 · doi:10.1017/CBO9780511542879
[3] Breuillard, Emmanuel; Varj\'u, P\'eter P., Entropy of {B}ernoulli convolutions and uniform exponential growth for linear groups, (2018) · Zbl 1442.37016
[4] Breuillard, Emmanuel; Varj\'u, P. P., On the dimension of {B}ernoulli convolutions, (2018) · Zbl 1429.28010
[5] Erd\"{o}s, Paul, On a family of symmetric {B}ernoulli convolutions, Amer. J. Math.. American Journal of Mathematics, 61, 974-976, (1939) · doi:10.2307/2371641
[6] Erd\"{o}s, Paul, On the smoothness properties of a family of {B}ernoulli convolutions, Amer. J. Math.. American Journal of Mathematics, 62, 180-186, (1940) · Zbl 0022.35403 · doi:10.2307/2371446
[7] Falconer, Kenneth, Techniques in Fractal Geometry, xviii+256 pp., (1997) · Zbl 0869.28003
[8] Feng, De-Jun; Hu, Huyi, Dimension theory of iterated function systems, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 62, 1435-1500, (2009) · Zbl 1230.37031 · doi:10.1002/cpa.20276
[9] Garsia, Adriano M., Arithmetic properties of {B}ernoulli convolutions, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 102, 409-432, (1962) · Zbl 0103.36502 · doi:10.2307/1993615
[10] Garsia, Adriano M., Entropy and singularity of infinite convolutions, Pacific J. Math.. Pacific Journal of Mathematics, 13, 1159-1169, (1963) · Zbl 0126.14901 · doi:10.2140/pjm.1963.13.1159
[11] Gou\`“ezel, S., M\'”ethodes entropiques pour les convolutions de {B}ernoulli, d’apr\`“es {H}ochman, {S}hmerkin, {B}reuillard, {V}arj\'”u, (2018)
[12] Hochman, Michael, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2). Annals of Mathematics. Second Series, 180, 773-822, (2014) · Zbl 1337.28015 · doi:10.4007/annals.2014.180.2.7
[13] Mignotte, Maurice, Approximation des nombres alg\'{e}briques par des nombres alg\'{e}briques de grand degr\'{e}, Ann. Fac. Sci. Toulouse Math. (5). Toulouse. Facult\'{e} des Sciences. Annales. Math\'{e}matiques. S\'{e}rie 5, 1, 165-170, (1979) · Zbl 0421.10022
[14] Peres, Yuval; Schlag, Wilhelm; Solomyak, Boris, Sixty years of {B}ernoulli convolutions. Fractal Geometry and Stochastics, {II}, Progr. Probab., 46, 39-65, (2000) · Zbl 0961.42006 · doi:10.1007/978-3-0348-8380-1_2
[15] Peres, Yuval; Solomyak, Boris, Absolute continuity of {B}ernoulli convolutions, a simple proof, Math. Res. Lett.. Mathematical Research Letters, 3, 231-239, (1996) · Zbl 0867.28001 · doi:10.4310/MRL.1996.v3.n2.a8
[16] Peres, Yuval; Solomyak, Boris, Problems on self-similar sets and self-affine sets: an update. Fractal Geometry and Stochastics, {II}, Progr. Probab., 46, 95-106, (2000) · Zbl 0946.28003 · doi:10.1007/978-3-0348-8380-1_4
[17] Pollicott, Mark; Simon, K\'{a}roly, The {H}ausdorff dimension of {\( \lambda \)}-expansions with deleted digits, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 347, 967-983, (1995) · Zbl 0831.28005 · doi:10.2307/2154881
[18] Shmerkin, Pablo, On the exceptional set for absolute continuity of {B}ernoulli convolutions, Geom. Funct. Anal.. Geometric and Functional Analysis, 24, 946-958, (2014) · Zbl 1305.28012 · doi:10.1007/s00039-014-0285-4
[19] Shmerkin, Pablo, On {F}urstenberg’s intersection conjecture, self-similar measures, and the {\(L^q\)} norms of convolutions, Ann. of Math. (2). Annals of Mathematics. Second Series, 189, 319-391, (2019) · Zbl 1426.11079 · doi:10.4007/annals.2019.189.2.1
[20] Solomyak, Boris, On the random series {\( \sum\pm\lambda^n\)} (an {E}rd{\H{o}}s problem), Ann. of Math. (2). Annals of Mathematics. Second Series, 142, 611-625, (1995) · Zbl 0837.28007 · doi:10.2307/2118556
[21] Solomyak, Boris, Notes on {B}ernoulli convolutions. Fractal Geometry and Applications: A Jubilee of {B}eno\^it {M}andelbrot. {P}art 1, Proc. Sympos. Pure Math., 72, 207-230, (2004) · Zbl 1115.28009 · doi:10.1090/pspum/072.1/2112107
[22] Varj\'{u}, P\'{e}ter P., Recent progress on {B}ernoulli convolutions · Zbl 1403.28010
[23] Varj\'{u}, P\'{e}ter P., Absolute continuity of {B}ernoulli convolutions for algebraic parameters, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 32, 351-397, (2019) · Zbl 1408.28017 · doi:10.1090/jams/916
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