Geodesic
Absolute continuity of Bernoulli convolutions for algebraic parameters
Varjú, Péter1
1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Journal of the American Mathematical Society, Tome 32 (2019) no. 2, pp. 351-397

Voir la notice de l'article provenant de la source American Mathematical Society

We prove that Bernoulli convolutions $\mu _\lambda$ are absolutely continuous provided the parameter $\lambda$ is an algebraic number sufficiently close to $1$ depending on the Mahler measure of $\lambda$.
DOI : 10.1090/jams/916

Varjú, Péter 1

1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 
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Varjú, Péter. Absolute continuity of Bernoulli convolutions for algebraic parameters. Journal of the American Mathematical Society, Tome 32 (2019) no. 2, pp. 351-397. doi: 10.1090/jams/916

[1] Bombieri, Enrico, Gubler, Walter Heights in Diophantine geometry 2006

[2] Bourgain, Jean The discretized sum-product and projection theorems J. Anal. Math. 2010 193 236

[3] Breuillard, E., Varjãº, P. P. On the dimension of Bernoulli convolutions, to appear in Ann. Probab.

[4] Breuillard, Emmanuel, Varjãº, Pã©Ter P. Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, to appear in J. Anal. Math.

[5] Cover, Thomas M., Thomas, Joy A. Elements of information theory 2006

[6] Dobrowolski, E. On a question of Lehmer and the number of irreducible factors of a polynomial Acta Arith. 1979 391 401

[7] Erdã¶S, Paul On a family of symmetric Bernoulli convolutions Amer. J. Math. 1939 974 976

[8] Erdã¶S, Paul On the smoothness properties of a family of Bernoulli convolutions Amer. J. Math. 1940 180 186

[9] Falconer, Kenneth Fractal geometry 2014

[10] Feng, De-Jun, Hu, Huyi Dimension theory of iterated function systems Comm. Pure Appl. Math. 2009 1435 1500

[11] Garsia, Adriano M. Arithmetic properties of Bernoulli convolutions Trans. Amer. Math. Soc. 1962 409 432

[12] Garsia, Adriano M. Entropy and singularity of infinite convolutions Pacific J. Math. 1963 1159 1169

[13] Hochman, Michael On self-similar sets with overlaps and inverse theorems for entropy Ann. of Math. (2) 2014 773 822

[14] Jessen, Bã¸Rge, Wintner, Aurel Distribution functions and the Riemann zeta function Trans. Amer. Math. Soc. 1935 48 88

[15] Kaä­Manovich, V. A., Vershik, A. M. Random walks on discrete groups: boundary and entropy Ann. Probab. 1983 457 490

[16] Kontoyiannis, Ioannis, Madiman, Mokshay Sumset and inverse sumset inequalities for differential entropy and mutual information IEEE Trans. Inform. Theory 2014 4503 4514

[17] Lindenstrauss, Elon, Varjãº, Pã©Ter P. Work in progress 2018

[18] Madiman, Mokshay On the entropy of sums in Information Theory Workshop, 2008, IEEE, 2008

[19] Peres, Yuval, Schlag, Wilhelm, Solomyak, Boris Sixty years of Bernoulli convolutions 2000 39 65

[20] Shmerkin, Pablo On the exceptional set for absolute continuity of Bernoulli convolutions Geom. Funct. Anal. 2014 946 958

[21] Smyth, Chris The Mahler measure of algebraic numbers: a survey 2008 322 349

[22] Solomyak, Boris On the random series ∑±𝜆ⁿ (an Erdős problem) Ann. of Math. (2) 1995 611 625

[23] Solomyak, Boris Notes on Bernoulli convolutions 2004 207 230

[24] Tao, Terence Sumset and inverse sumset theory for Shannon entropy Combin. Probab. Comput. 2010 603 639

[25] Varjãº, Pã©Ter P. Recent progress on Bernoulli convolutions

[26] Wang, Zhiren Quantitative density under higher rank abelian algebraic toral actions Int. Math. Res. Not. IMRN 2011 3744 3821

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