Fourier decay for self-similar measures

Solomyak, Boris

doi:10.1090/proc/15515

Fourier decay for self-similar measures
HTML articles powered by AMS MathViewer

by Boris Solomyak PDF

Proc. Amer. Math. Soc. 149 (2021), 3277-3291 Request permission

Abstract:

We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erdős and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.

References

Noga Alon and Joel H. Spencer, The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1992. With an appendix by Paul Erdős; A Wiley-Interscience Publication. MR 1140703
Jean Bourgain and Semyon Dyatlov, Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal. 27 (2017), no. 4, 744–771. MR 3678500, DOI 10.1007/s00039-017-0412-0
Alexander I. Bufetov and Boris Solomyak, On the modulus of continuity for spectral measures in substitution dynamics, Adv. Math. 260 (2014), 84–129. MR 3209350, DOI 10.1016/j.aim.2014.04.004
Alexander I. Bufetov and Boris Solomyak, The Hölder property for the spectrum of translation flows in genus two, Israel J. Math. 223 (2018), no. 1, 205–259. MR 3773061, DOI 10.1007/s11856-017-1614-8
Fan Chung and Linyuan Lu, Concentration inequalities and martingale inequalities: a survey, Internet Math. 3 (2006), no. 1, 79–127. MR 2283885, DOI 10.1080/15427951.2006.10129115
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
H. Davenport, P. Erdős, and W. J. LeVeque, On Weyl’s criterion for uniform distribution, Michigan Math. J. 10 (1963), 311–314. MR 153656, DOI 10.1307/mmj/1028998917
Paul Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974–976. MR 311, DOI 10.2307/2371641
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
Kenneth J. Falconer and Xiong Jin, Exact dimensionality and projections of random self-similar measures and sets, J. Lond. Math. Soc. (2) 90 (2014), no. 2, 388–412. MR 3263957, DOI 10.1112/jlms/jdu031
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
Børge Jessen and Aurel Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88. MR 1501802, DOI 10.1090/S0002-9947-1935-1501802-5
Thomas Jordan and Tuomas Sahlsten, Fourier transforms of Gibbs measures for the Gauss map, Math. Ann. 364 (2016), no. 3-4, 983–1023. MR 3466857, DOI 10.1007/s00208-015-1241-9
Antti Käenmäki and Tuomas Orponen, Absolute continuity in families of parametrised non-homogeneous self-similar measures, arXiv e-prints (2018), arXiv:1812.05006.
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
Robert Kaufman, On Bernoulli convolutions, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 217–222. MR 737403, DOI 10.1090/conm/026/737403
Jialun Li, Decrease of Fourier coefficients of stationary measures, Math. Ann. 372 (2018), no. 3-4, 1189–1238. MR 3880297, DOI 10.1007/s00208-018-1743-3
Jialun Li, Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups, arXiv e-prints (2018), arXiv:1811.06484.
Jialun Li, Frederic Naud, and Wenyu Pan, Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay, arXiv e-prints (2019), arXiv:1902.01103.
Jialun Li and Tuomas Sahlsten, Trigonometric series and self-similar sets, arXiv e-prints (2019), arXiv:1902.00426.
Jialun Li and Tuomas Sahlsten, Fourier transform of self-affine measures, Adv. Math. 374 (2020), 107349, 35. MR 4133521, DOI 10.1016/j.aim.2020.107349
Russell Lyons, Seventy years of Rajchman measures, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), 1995, pp. 363–377 (English, with English and French summaries). MR 1364897
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
Carolina A. Mosquera and Pablo S. Shmerkin, Self-similar measures: asymptotic bounds for the dimension and Fourier decay of smooth images, Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 2, 823–834. MR 3839838, DOI 10.5186/aasfm.2018.4350
J. Neunhäuserer, Properties of some overlapping self-similar and some self-affine measures, Acta Math. Hungar. 92 (2001), no. 1-2, 143–161. MR 1924256, DOI 10.1023/A:1013716430425
Sze-Man Ngai and Yang Wang, Self-similar measures associated to IFS with non-uniform contraction ratios, Asian J. Math. 9 (2005), no. 2, 227–244. MR 2176605, DOI 10.4310/AJM.2005.v9.n2.a7
Yuval Peres, Wilhelm Schlag, and Boris Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 39–65. MR 1785620
Martine Queffélec and Olivier Ramaré, Analyse de Fourier des fractions continues à quotients restreints, Enseign. Math. (2) 49 (2003), no. 3-4, 335–356 (French, with English summary). MR 2028020
Santiago Saglietti, Pablo Shmerkin, and Boris Solomyak, Absolute continuity of non-homogeneous self-similar measures, Adv. Math. 335 (2018), 60–110. MR 3836658, DOI 10.1016/j.aim.2018.06.015
Tuomas Sahlsten and Connor Stevens, Fourier decay in nonlinear dynamics, arXiv e-prints (2018), arXiv:1810.01378.
R. Salem, Sets of uniqueness and sets of multiplicity, Trans. Amer. Math. Soc. 54 (1943), 218–228. MR 8428, DOI 10.1090/S0002-9947-1943-0008428-8
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
Pablo Shmerkin and Boris Solomyak, Absolute continuity of complex Bernoulli convolutions, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 3, 435–453. MR 3569155, DOI 10.1017/S0305004116000335
Pablo Shmerkin and Boris Solomyak, Absolute continuity of self-similar measures, their projections and convolutions, Trans. Amer. Math. Soc. 368 (2016), no. 7, 5125–5151. MR 3456174, DOI 10.1090/tran6696
Masato Tsujii, On the Fourier transforms of self-similar measures, Dyn. Syst. 30 (2015), no. 4, 468–484. MR 3430311, DOI 10.1080/14689367.2015.1078291
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
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