Self-similar measures and the Rajchman property
Annales Henri Lebesgue, Volume 4 (2021) , pp. 973-1004.

Metadata

KeywordsRajchman measure, self-similar measure, Pisot number, Plastic number

Abstract

For general self-similar measures associated with contracting on average affine IFS on the real line, we study the convergence to zero of the Fourier transform at infinity (or Rajchman property) and the extension of results of Salem [Sal44] and Erdös [Erd39] on Bernoulli convolutions. Revisiting in a first step a recent work of Li–Sahlsten [LS19], we show that the parameters where the Rajchman property may not hold are very special and in close connection with Pisot numbers. In these particular cases, the Rajchman character appears to be equivalent to absolute continuity and, when the IFS consists of strict contractions, we show that it is generically not true. We finally provide rather surprising numerical simulations and an application to sets of multiplicity for trigonometric series.


References

[Bal00] Baladi, Viviane Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific, 2000 | MR 1793194 | Zbl 1012.37015

[Blu00] Bluhm, Christian E. Liouville numbers, Rajchman measures, and small Cantor sets, Proc. Am. Math. Soc., Volume 128 (2000) no. 9, pp. 2637-2640 | Article | MR 1657762 | Zbl 1003.11036

[Cas57] Cassels, J. W. S. An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, 45, Cambridge University Press, 1957 | MR 87708 | Zbl 0077.04801

[Erd39] Erdös, Paul On a family of symmetric Bernoulli convolutions, Am. J. Math., Volume 61 (1939), pp. 974-975 | Article | MR 311 | Zbl 0022.35042

[Fal03] Falconer, Kenneth Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 2003 | Zbl 1060.280056

[Hoc14] Hochman, Michael On self-similar sets with overlaps and inverse theorems for entropy, Ann. Math., Volume 180 (2014) no. 2, pp. 773-822 | Article | MR 3224722 | Zbl 1337.28015

[Hut81] Hutchinson, John E. Fractals and self-similarity, Indiana Univ. Math. J., Volume 30 (1981), pp. 713-747 | Article | MR 625600 | Zbl 0598.28011

[JR08] Jaroszewska, Joanna; Rams, Michal On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS, J. Stat. Phys., Volume 132 (2008) no. 5, pp. 907-919 | Article | MR 2430786 | Zbl 1158.28003

[Kat76] Kato, Tosio Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, 1976 | MR 407617 | Zbl 0342.47009

[KP92] Krantz, Steven; Parks, Harold R. A primer of real analytic functions, Birkhäuser, 1992 | Zbl 0767.26001

[LS19] Li, Jialun; Sahlsten, Tuomas Trigonometric series and self-similar sets (2019) (https://arxiv.org/abs/1902.00426)

[Lyo95] Lyons, Russell Seventy years of Rajchman measures, J. Fourier Anal. Appl., Volume Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), Special Issue (1995), pp. 363-377 | MR 1364897 | Zbl 0886.43001

[Men16] Menshov, Dmitriĭ E. Sur l’unicité du développement trigonométrique, C. R. Math. Acad. Sci. Paris, Volume 163 (1916), pp. 433-436 | Zbl 46.0457.02

[Neu11] Neunhäuserer, Jörg A family of exceptional parameters for non-uniform self-similar measures, Electron. Commun. Probab., Volume 16 (2011), pp. 192-199 | MR 2788891 | Zbl 1223.28005

[PSS00] Peres, Yuval; Schlag, Wilhelm; Solomyak, Boris Sixty years of Bernoulli convolutions, Fractal geometry and stochastics II. Proceedings of the 2nd conference, Greifswald/ Koserow, Germany, August 28-September 2, 1998 (Progress in Probability), Volume 46 (2000), pp. 39-65 | MR 1785620 | Zbl 0961.42006

[Ros55] Rosenbloom, Paul Perturbation of linear operators in Banach space, Arch. Math., Volume 6 (1955), pp. 89-101 | Article | MR 68118 | Zbl 0065.10502

[Sal44] Salem, Raphael A remarkable class of algebraic integers. Proof of a conjecture by Vijayaraghavan, Duke Math. J., Volume 11 (1944), pp. 103-108 | Zbl 0063.06657

[Sal63] Salem, Raphael Algebraic numbers and Fourier analysis, D.C. Heath & Co’s. publications, 1963 | MR 157941 | Zbl 0126.07802

[Sam70] Samuel, Pierre Algebraic theory of numbers, Houghton Mifflin Co., Boston, 1970 (Translated from the French by Allan J. Silberger) | Zbl 0215.36001

[Sie44] Siegel, Carl. L. Algebraic numbers whose conjugates lie in the unit circle, Duke Math. J., Volume 11 (1944), pp. 597-602 | Zbl 0063.07005

[Sol04] Solomyak, Boris Notes on Bernoulli convolutions, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1: Analysis, number theory, and dynamical systems (Proceedings of Symposia in Pure Mathematics), Volume 72 (2004), pp. 207-230 | MR 2112107 | Zbl 1115.28009

[Sol19] Solomyak, Boris Fourier decay for self-similar measures (2019) (https://arxiv.org/abs/1906.12164) | Zbl 07357556

[Spi76] Spitzer, Frank Principles of random walks, Graduate Texts in Mathematics, 34, Springer, 1976 | MR 388547 | Zbl 0359.60003

[Str90] Strichartz, Robert S. Self-Similar Measures and Their Fourier Transforms I, Indiana Univ. Math. J., Volume 39 (1990) no. 3, pp. 797-817 | Article | MR 1078738 | Zbl 0695.28003

[Str93] Strichartz, Robert S. Self-Similar Measures and Their Fourier Transforms II, Trans. Am. Math. Soc., Volume 336 (1993) no. 1, pp. 335-361 | Article | MR 1081941 | Zbl 0765.28007

[Tsu15] Tsujii, Masato On the Fourier transforms of self-similar measures, Dyn. Syst., Volume 30 (2015) no. 4, pp. 468-484 | Article | MR 3430311 | Zbl 1329.28022

[VY20] Varju, Péter; Yu, Han Fourier decay of self-similar measures and self-similar sets of uniqueness (2020) (https://arxiv.org/abs/2004.09358)

[Woo82] Woodroofe, Michael Nonlinear renewal theory in sequential analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 39, Society for Industrial and Applied Mathematics, 1982 | MR 660065 | Zbl 0487.62062

[Zyg59] Zygmund, Antoni Trigonometric series. Vol. I and II, 1959, Cambridge University Press, 1959 | Zbl 0085.05601