Bernoulli convolutions with Garsia parameters in $(1,\sqrt {2}]$ have continuous density functions
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Abstract:
Let $\lambda \in (1,\sqrt {2}]$ be an algebraic integer with Mahler measure $2$. A classical result of Garsia shows that the Bernoulli convolution $\mu _\lambda$ is absolutely continuous with respect to the Lebesgue measure with a density function in $L^\infty$. In this paper, we show that the density function is continuous.References
- Shigeki Akiyama, De-Jun Feng, Tom Kempton, and Tomas Persson, On the Hausdorff dimension of Bernoulli convolutions, Int. Math. Res. Not. IMRN 19 (2020), 6569–6595. MR 4165484, DOI 10.1093/imrn/rny209
- Emmanuel Breuillard and Péter P. Varjú, On the dimension of Bernoulli convolutions, Ann. Probab. 47 (2019), no. 4, 2582–2617. MR 3980929, DOI 10.1214/18-AOP1324
- Emmanuel Breuillard and Péter P. Varjú, Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, J. Anal. Math. 140 (2020), no. 2, 443–481. MR 4093919, DOI 10.1007/s11854-020-0100-0
- Xin-Rong Dai, De-Jun Feng, and Yang Wang, Refinable functions with non-integer dilations, J. Funct. Anal. 250 (2007), no. 1, 1–20. MR 2345903, DOI 10.1016/j.jfa.2007.02.005
- Paul Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180–186. MR 858, DOI 10.2307/2371446
- De-Jun Feng and Huyi Hu, Dimension theory of iterated function systems, Comm. Pure Appl. Math. 62 (2009), no. 11, 1435–1500. MR 2560042, DOI 10.1002/cpa.20276
- 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
- Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316, DOI 10.1007/978-0-387-09434-2
- Kevin G. Hare and Maysum Panju, Some comments on Garsia numbers, Math. Comp. 82 (2013), no. 282, 1197–1221. MR 3008855, DOI 10.1090/S0025-5718-2012-02636-6
- Yanick Heurteaux, Dimension of measures: the probabilistic approach, Publ. Mat. 51 (2007), no. 2, 243–290. MR 2334791, DOI 10.5565/PUBLMAT_{5}1207_{0}1
- Michael Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180 (2014), no. 2, 773–822. MR 3224722, DOI 10.4007/annals.2014.180.2.7
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
- J.-P. Kahane, Sur la distribution de certaines séries aléatoires, Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Bull. Soc. Math. France, Mém. No. 25, Soc. Math. France Paris, 1971, pp. 119–122 (French). MR 0360498, DOI 10.24033/msmf.42
- S. Kittle, Absolute continuity of self similar measures, arXiv:2103.12684, 2020.
- Pertti Mattila, Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, vol. 150, Cambridge University Press, Cambridge, 2015. MR 3617376, DOI 10.1017/CBO9781316227619
- Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193–251. MR 1749437, DOI 10.1215/S0012-7094-00-10222-0
- 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
- Pablo Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal. 24 (2014), no. 3, 946–958. MR 3213835, DOI 10.1007/s00039-014-0285-4
- Pablo Shmerkin, On Furstenberg’s intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, Ann. of Math. (2) 189 (2019), no. 2, 319–391. MR 3919361, DOI 10.4007/annals.2019.189.2.1
- Péter P. Varjú, Absolute continuity of Bernoulli convolutions for algebraic parameters, J. Amer. Math. Soc. 32 (2019), no. 2, 351–397. MR 3904156, DOI 10.1090/jams/916
- Péter P. Varjú, On the dimension of Bernoulli convolutions for all transcendental parameters, Ann. of Math. (2) 189 (2019), no. 3, 1001–1011. MR 3961088, DOI 10.4007/annals.2019.189.3.9
Additional Information
- Han Yu
- Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 1223262
- Email: hy351@cam.ac.uk
- Received by editor(s): September 3, 2021
- Received by editor(s) in revised form: November 22, 2021, November 23, 2021, and December 29, 2021
- Published electronically: May 27, 2022
- Additional Notes: The author was financially supported by the University of Cambridge and the Corpus Christi College, Cambridge. The author had received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 803711)
- Communicated by: Katrin Gelfert
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 4359-4368
- MSC (2020): Primary 28A78, 42A85, 37A44
- DOI: https://doi.org/10.1090/proc/15971
- MathSciNet review: 4470180