Self-similar measures and the Rajchman property

Brémont, Julien

Annales Henri Lebesgue, Tome 4 (2021), pp. 973-1004.

Pour des mesures auto-similaires générales associées à des IFS affines et contractant en moyenne, nous étudions la convergence vers zéro à l’infini de la transformée de Fourier (propriété de Rajchman) et l’extension de résultats de Salem [Sal44] et Erdös [Erd39] sur les convolutions de Bernoulli. Reprenant dans une première étape des travaux récents de Li–Sahlsten [LS19], nous montrons que les paramètres où la propriété de Rajchman pourrait ne pas être vraie sont très spéciaux et en lien étroit avec les nombres de Pisot. Dans ces cas particuliers, la propriété de Rajchman s’avère être équivalente à l’absolue continuité et, quand l’IFS est constitué de contractions strictes, nous montrons qu’elle est génériquement fausse. Nous terminons ce travail par d’assez surprenantes simulations numériques et une application aux ensembles de multiplicité pour les séries trigonométriques.

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.

Classification : 11K16, 37A45, 42A38, 42A61, 60K20
Mots clés : Rajchman measure, self-similar measure, Pisot number, Plastic number

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

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

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

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

[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, Birkhäuser (2000), pp. 39-65 | MR | Zbl

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

[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

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

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

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

[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, American Mathematical Society (2004), pp. 207-230 | MR | Zbl

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

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

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

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

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

[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 | Zbl

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