Abstract
Let Ë be a small subset of a finite abelian group, and let R be the set of points at which its Fourier transform is large. A result of Chang states that R has a great deal of additive structure. We give a statement and a proof of this result and discuss some applications of it. Finally, we discuss some related open questions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ball, K., Convex geometry and functional analysis, in Handbook of the Geometry of Ba-nach Spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 161–194.
Beckner, W., Inequalities in Fourier analysis, Ann. Math. 102 no. 1 (1975), 159–182.
Bourgain, J., Sur le minimum d’une somme de cosinus, Acta Arith. 45, no. 4 (1986), 381–389.
Bourgain, J., On arithmetic progressions in sums of sets of integers, in A Tribute to Paul Erdós Cambridge University Press, London, 1990.
Chang, M.C., Polynomial bounds for Freimans’s theorem, Duke Math. J. 113 no. 3 (2002), 399–419.
de Leeuw, K., Katznelson, Y., and Kahane, J.P., Sur les coefficients de Fourier des fonc-tions continues, CR. Acad. Sei. Pans Sér. A-B 285 no. 16 (1977), A1001–A1003.
Green, B.J., Arithmetic progressions in sumsets, GAFA 12 (2002) 584–597.
Green, B.J. Finite field models in additive number theory, to appear in Surveys in Combinatorics 2005.
Green, B.J., Some constructions in the inverse spectral theory of cyclic groups, Comb. Prob. Comp. 12 no. 2 (2003), 127–138.
Green, B.J.Various unpublished lecture notes1. Large deviation inequalities2. Edinburgh lecture notes on Freiman’s theorem3. Restriction and Kakeya phenomena (Part III course, Cambridge 2002)
Konyagin, S.V., On a problem of Littlewood, Izv. Akad. Nauk. SSSR 45 (1981), 243–265 (in Russian); English trans.:Math. USSR-Izv 18 (1982), 205–225
López, J.M. and Ross, K.A., Sidon Sets Lecture Notes in Pure and Applied Mathematics, Vol. 13, Marcel Dekker, Inc., New York, 1975.
McGehee, O.C., Pigno, L., and Smith, B., Hardy’s inequality and the L1 -norm of expo-nential sums, Ann. Math 113 no. 3 (1981), 613–618
Nazarov, F.L., The Bang solution of the coefficient problem, Algebra i Analiz. 9, no. 2 (1997), 272–287; English translation in St. Petersburg Math. J. 9 no. 2 (1998), 407–419
Rudin, W., Fourier Analysis on Groups Wiley 1990 (reprint of the 1962 original)
Ruzsa, I.Z., Arithmetic progressions in sumsets,Acta Arith 60no. 2 (1991), 191–202
Ruzsa, I.Z., Connections between the uniform distribution of a sequence and its differences, in Topics in Classical Number Theory Vol. I, II (Budapest, 1981), 1419–1443,Colloq. Math. Soc. János Bolyai, 34, North-Holland, Amsterdam, 1984
Ruzsa, I.Z., An analog of Freiman’s theorem in groups, Structure theory of set addition, Ast¨¦risque 258 (1999), 323–326.
Ruzsa, I.Z., Negative values of cosine sums, Acta Arith 111 (2004),179–186
Spencer, J., Six standard deviations suffice, Trans. Amer. Math. Soc 289 no. 2 (1985), 679–706
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media New York
About this chapter
Cite this chapter
Green, B. (2004). Spectral Structure of Sets of Integers. In: Brandolini, L., Colzani, L., Travaglini, G., Iosevich, A. (eds) Fourier Analysis and Convexity. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8172-2_4
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8172-2_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6474-3
Online ISBN: 978-0-8176-8172-2
eBook Packages: Springer Book Archive