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

Hare, Kathryn; Roginskaya, Maria

doi:10.1090/S0002-9939-02-06826-0

A Fourier series formula for energy of measures with applications to Riesz products
HTML articles powered by AMS MathViewer

by Kathryn E. Hare and Maria Roginskaya PDF

Proc. Amer. Math. Soc. 131 (2003), 165-174 Request permission

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.

References

Jean Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), no. 1-3, 193–201. MR 1286826, DOI 10.1007/BF02772994
Gavin Brown, William Moran, and Charles E. M. Pearce, Riesz products, Hausdorff dimension and normal numbers, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 529–540. MR 878900, DOI 10.1017/S0305004100066895
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
Fan Ai Hua, Quelques propriétés des produits de Riesz, Bull. Sci. Math. 117 (1993), no. 4, 421–439 (French, with English and French summaries). MR 1245805
Ai Hua Fan, Une formule approximative de dimension pour certains produits de Riesz, Monatsh. Math. 118 (1994), no. 1-2, 83–89 (French, with English summary). MR 1289850, DOI 10.1007/BF01305776
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Pertti Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34 (1987), no. 2, 207–228. MR 933500, DOI 10.1112/S0025579300013462
Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193–251. MR 1749437, DOI 10.1215/S0012-7094-00-10222-0
Jacques Peyrière, Étude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 2, xii, 127–169. MR 404973
Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227. MR 0116177, DOI 10.1512/iumj.1960.9.59013
Per Sjölin and Fernando Soria, Some remarks on restriction of the Fourier transform for general measures, Publ. Mat. 43 (1999), no. 2, 655–664. MR 1744623, DOI 10.5565/PUBLMAT_{4}3299_{0}7
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776