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On the Fourier dimension and a modification. (English) Zbl 1327.42010

Summary: We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally, we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that for each \(s\) there is a class of sets such that a measure has modified Fourier dimension greater than or equal to \(s\) if and only if it annihilates all sets in the class.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
28A80 Fractals
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References:

[1] A. Besicovitch, Sets of fractional dimension. IV. On rational approximation to real numbers. J. London Math. Soc. 9 (1934), 126-131. · Zbl 0009.05301 · doi:10.1112/jlms/s1-9.2.126
[2] Ch. Bluhm, On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat. 36 (1998), no. 2, 307-316. · Zbl 1026.28004 · doi:10.1007/BF02384771
[3] H. Davenport, P. Erdős, and W. J. LeVeque, On Weyl’s criterion for uniform distri- bution. Michigan Math. J. 10 (1963), 311-314. · Zbl 0119.28201 · doi:10.1307/mmj/1028998917
[4] W. L. Fouché and S. Mukeru, On the Fourier structure of the zero set of frac- tional Brownian motion. Stat. and Probab. Lett. 83 (2013), 459-466. · Zbl 1267.60041 · doi:10.1016/j.spl.2012.10.015
[5] J. M. Fraser, T. Orponen, and T. Sahlsten, On Fourier analytic properties of graphs. Int. Math. Res. Not. IMRN 2014 (2014), no. 10, 2730-2745. · Zbl 1305.60028 · doi:10.1093/imrn/rnt015
[6] A. Goullet de Rugy, Sur les mesures étrangères. C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A123-A126. · Zbl 0202.33601
[7] V. Jarník, Diophantischen Approximationen und Hausdorffsches Maß. Matematich- eskii Sbornik (3-4) 36 (1929), 371-382.
[8] Th. Jordan and T. Sahlsten, Fourier transforms of Gibbs measures for the Gauss map. Preprint 2013. · Zbl 1343.42006
[9] J.-P. Kahane, Some random series of functions. 2nd ed. Cambridge Studies in Ad- vanced Mathematics, 5. Cambridge University Press, Cambridge, 1985, Chapters 17 and 18. · Zbl 0571.60002
[10] R. Kaufman, Random measures on planar curves. Ark. Mat. 14 (1976), no. 2, 245-250. · Zbl 0371.60026 · doi:10.1007/BF02385838
[11] R. Kaufman, On the theorem of Jarník and Besicovitch. Acta Arith. 39 (1981), no. 3, 265-267. · Zbl 0468.10031
[12] R. Lyons, Seventy years of Rajchman measures. J. Fourier Anal. Appl. (1995), Spe- cial Issue, 363-377. Proceedings of the Conference in Honor of J.-P. Kahane (Orsay, 1993). · Zbl 0886.43001
[13] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifia- bility. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. · Zbl 0819.28004
[14] J. C. Oxtoby, Measure and category. A survey of the analogies between topological and measure spaces. 2nd ed. Graduate Texts in Mathematics, 2. Springer, Berlin etc., 1980. · Zbl 0435.28011
[15] M. Queffélec and O. Ramaré, Analyse de Fourier des fractions continues à quo- tients restreints. Enseign. Math. (2) 49 (2003), no. 3-4, 335-356. · Zbl 1057.11038
[16] C. A. Rogers, Hausdorff measures. Cambridge University Press, London and New York, 1970. · Zbl 0204.37601
[17] R. Salem, On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1 (1951), no. 4, 353-365. · Zbl 0054.03001 · doi:10.1007/BF02591372
[18] Th. H. Wolff, Lectures on harmonic analysis. With a foreword by Ch. Fefferman and preface by I. Łaba. Edited by I. Łaba and C. Shubin. University Lecture Se- ries, 29. American Mathematical Society, Providence, R.I., 2003. · Zbl 1041.42001
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