On a theorem of Saeki concerning convolution squares of singular measures

Thomas Körner

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 3, page 439-464
  • ISSN: 0037-9484

Abstract

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If 1 > α > 1 / 2 , then there exists a probability measure μ such that the Hausdorff dimension of the support of μ is α and μ * μ is a Lipschitz function of class α - 1 2 .

How to cite

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Körner, Thomas. "On a theorem of Saeki concerning convolution squares of singular measures." Bulletin de la Société Mathématique de France 136.3 (2008): 439-464. <http://eudml.org/doc/272433>.

@article{Körner2008,
abstract = {If $1&gt;\alpha &gt;1/2$, then there exists a probability measure $\mu $ such that the Hausdorff dimension of the support of $\mu $ is $\alpha $ and $\mu *\mu $ is a Lipschitz function of class $\alpha -\tfrac\{1\}\{2\}$.},
author = {Körner, Thomas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {convolution square; self convolution; singular measure},
language = {eng},
number = {3},
pages = {439-464},
publisher = {Société mathématique de France},
title = {On a theorem of Saeki concerning convolution squares of singular measures},
url = {http://eudml.org/doc/272433},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Körner, Thomas
TI - On a theorem of Saeki concerning convolution squares of singular measures
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 3
SP - 439
EP - 464
AB - If $1&gt;\alpha &gt;1/2$, then there exists a probability measure $\mu $ such that the Hausdorff dimension of the support of $\mu $ is $\alpha $ and $\mu *\mu $ is a Lipschitz function of class $\alpha -\tfrac{1}{2}$.
LA - eng
KW - convolution square; self convolution; singular measure
UR - http://eudml.org/doc/272433
ER -

References

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  1. [1] N. K. Bary – A treatise on trigonometric series. Vols. I, II, Authorized translation by Margaret F. Mullins. A Pergamon Press Book, The Macmillan Co., 1964. Zbl0129.28002MR171116
  2. [2] C. C. Graham & O. C. McGehee – Essays in commutative harmonic analysis, Grund. Math. Wiss., vol. 238, Springer, 1979. Zbl0439.43001MR550606
  3. [3] G. R. Grimmett & D. R. Stirzaker – Probability and random processes, third éd., Oxford University Press, 2001. Zbl0759.60001MR2059709
  4. [4] S. K. Gupta & K. E. Hare – « On convolution squares of singular measures », Colloq. Math.100 (2004), p. 9–16. Zbl1052.43001MR2079343
  5. [5] F. Hausdorff – Set theory, Chelsea Publishing Company, New York, 1957. Zbl0081.04601MR86020
  6. [6] W. Hoeffding – « Probability inequalities for sums of bounded random variables », J. Amer. Statist. Assoc.58 (1963), p. 13–30. Zbl0127.10602MR144363
  7. [7] J.-P. Kahane & R. Salem – Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust., No. 1301, Hermann, 1963. Zbl0112.29304MR160065
  8. [8] R. Kaufman – « Small subsets of finite abelian groups », Ann. Inst. Fourier (Grenoble) 18 (1968), p. 99–102 V. Zbl0175.30501MR241532
  9. [9] K. Kuratowski – Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, 1966. Zbl0158.40802MR217751
  10. [10] S. Saeki – « On convolution squares of singular measures », Illinois J. Math.24 (1980), p. 225–232. Zbl0496.42006MR575063
  11. [11] N. Wiener & A. Wintner – « Fourier-Stieltjes Transforms and Singular Infinite Convolutions », Amer. J. Math.60 (1938), p. 513–522. Zbl0019.16901MR1507332JFM64.0223.02

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