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A Fourier series formula for energy of measures with applications to Riesz products

Authors: Kathryn E. Hare and Maria Roginskaya
Journal: Proc. Amer. Math. Soc. 131 (2003), 165-174
MSC (2000): Primary 28A12, 42A55
Published electronically: June 12, 2002
MathSciNet review: 1929036
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Abstract: In this paper we derive a formula relating the energy and the Fourier transform of a finite measure on the $d$-dimensional torus which is similar to the well-known formula for measures on $\mathbb{R} ^{d}$.

We apply the formula to obtain estimates on the Hausdorff dimension of Riesz product measures. These give improvements on the earlier, classical results which were based on completely different techniques.

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Additional Information

Kathryn E. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Maria Roginskaya
Affiliation: Department of Mathematics, Chalmers TH and Goteborg University, Eklandagatan 86, SE-41296, Sweden

Keywords: energy, Hausdorff dimension, Riesz products
Received by editor(s): August 17, 2001
Published electronically: June 12, 2002
Additional Notes: This research was done while the first author enjoyed the hospitality of the Department of Mathematics at Göteborg University and Chalmers Institute of Technology. It was supported in part by NSERC and the Swedish Natural Sciences Research Council.
Communicated by: David Preiss
Article copyright: © Copyright 2002 American Mathematical Society