Skip to main content
SpringerLink
Log in

Coset decision trees and the Fourier algebra

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We show that if G is a finite group and f is a {0, 1}-valued function on G with Fourier algebra norm at most M, then f may be computed by the coset decision tree (that is a decision tree in which at each vertex we query membership of a given coset) having at most (exp(exp(exp(O(M2))))) leaves. A short calculation shows that any {0, 1}-valued function which may be computed by a coset decision tree with m leaves has Fourier algebra norm at most exp(O(m)).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Akavia, Finding significant Fourier transform coefficients deterministically and locally, Electron. Colloquium Comput. Complex. 102 (2008), 27. https://eccc.weizmann.ac.il/report/2008/102/download

    Google Scholar 

  2. A. Akavia, Deterministic sparse Fourier approximation via fooling arithmetic progressions, in COLT 2010—The 23rd Conference on Learning Theory, Haifa, Israel, June 27–29, 2010, Omnipress, 2010, pp. 381–393.

  3. A. Akavia, Deterministic sparse Fourier approximation via approximating arithmetic progressions, IEEE Trans. Inform. Theory 60 (2014), 1733–1741.

    Article  MathSciNet  Google Scholar 

  4. A. Akavia, Explicit small sets with ε-discrepancy on Bohr sets, Inform. Process. Lett. 114 (2014), 564–567.

    Article  MathSciNet  Google Scholar 

  5. E. Breuillard, B. J. Green and T. C. Tao, The structure appproximate groups, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221.

    Article  MathSciNet  Google Scholar 

  6. D. Boneh, Learning using group representations (extended abstract), in Proceedings of the Eighth Annual Conference on Computational Learning Theory, ACM, New York, 1995, pp. 418–426.

    Chapter  Google Scholar 

  7. E. S. Croot, I. Łaba and O. Sisask, Arithmetic progressions in sumsets and Lp-almost-periodicity. Combin. Probab. Comput. 22 (2013), 351–365.

    Article  MathSciNet  Google Scholar 

  8. P. J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191–212.

    Article  MathSciNet  Google Scholar 

  9. E. S. Croot and O. Sisask, A probabilistic technique for finding almost-periods of convolutions, Geom. Funct. Anal. 20 (2010), 1367–1396.

    Article  MathSciNet  Google Scholar 

  10. A. Czuron and M. Wojciechowski, On the isomorphisms of Fourier algebras of finite Abelian groups, arXiv:1306.1480 [math.FA]

  11. P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236.

    Article  MathSciNet  Google Scholar 

  12. L. Fejér, Lebesguessche Konstanten und divergente Fourierreihen, J. Reine Angew. Math. 138 (1910), 22–53.

    Article  MathSciNet  Google Scholar 

  13. B. Forrest, E. Kaniuth, A. T. Lau and N. Spronk, Ideals with bounded approximate identities in Fourier algebras, J. Funct. Anal. 203 (2003), 286–304.

    Article  MathSciNet  Google Scholar 

  14. B. E. Forrest and V. Runde, Amenability and weak amenability of the Fourier algebra, Math. Z. 250 (2005), 731–744.

    Article  MathSciNet  Google Scholar 

  15. B. J. Green and S. V. Konyagin, On the Littlewood problem modulo a prime, Canad. J. Math. 61 (2009), 141–164.

    Article  MathSciNet  Google Scholar 

  16. B. J. Green and T. Sanders, Boolean functions with small spectral norm, Geom. Funct. Anal. 18 (2008), 144–162.

    Article  MathSciNet  Google Scholar 

  17. B. J. Green and T. Sanders, A quantitative version of the idempotent theorem in harmonic analysis, Ann. of Math. (2) 168 (2008), 1025–1054.

    Article  MathSciNet  Google Scholar 

  18. B. Host, Le théorème des idempotents dans B(G), Bull. Soc. Math. France 114 (1986), 215–223.

    Article  MathSciNet  Google Scholar 

  19. E. Kushilevitz and Y. Mansour, Learning decision trees using the Fourier spectrum. SIAM J. Comput. 22 (1993), 1331–1348.

    Article  MathSciNet  Google Scholar 

  20. S. V. Konyagin and I. D. Shkredov, On the Wiener norm of subsets of ℤ/pℤ of medium size, J. Math. Sci. (N.Y.) 218 (2016), 599–608.

    Article  MathSciNet  Google Scholar 

  21. M. Lefranc, Sur certaines algèbres de fonctions sur un groupe, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1882–A1883.

    MathSciNet  MATH  Google Scholar 

  22. J.-F. Méla, Mesures ε-idempotentes de norme bornée, Studia Math. 72 (1982), 131–149.

    Article  MathSciNet  Google Scholar 

  23. R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, Cambridge, 2014.

    Book  Google Scholar 

  24. W. Rudin. Fourier Analysis on Groups, John Wiley & Sons, New York, 1990.

    Book  Google Scholar 

  25. V. Runde, Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca-Herz algebras, J. Math. Anal. Appl. 329 (2007), 736–751.

    Article  MathSciNet  Google Scholar 

  26. T. Sanders, A quantitative version of the non-abelian idempotent theorem. Geom. Funct. Anal. 21 (2011), 141–221.

    Article  MathSciNet  Google Scholar 

  27. T. Sanders, On the Bogolyubov-Ruzsa lemma, Anal. PDE 5 (2012), 627–655.

    Article  MathSciNet  Google Scholar 

  28. A. Shpilka, A. Tal and B. lee Volk, On the structure of Boolean functions with small spectral norm, Comput. Complexity 26 (2017), 229–273.

    Article  MathSciNet  Google Scholar 

  29. T. C. Tao and V. H. Vu, Additive Combinatorics, Cambridge University Press, Cambridge, 2006.

    Book  Google Scholar 

  30. P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press Cambridge, 1991.

    Book  Google Scholar 

  31. M. Wojciechowskiw The non-equivalence between the trigonometric system and the system of functions with pointwise restrictions on values in the uniform and L1norms, Math. Proc. Cambridge Philos. Soc. 150 (2011), 561–571.

    Article  MathSciNet  Google Scholar 

  32. D. Zwillinger, S. G. Krantz and K. H. Rosen, editors., CRC Standard Mathematical Tables and Formulae, CRC Press, Boca Raton, FL, 2003.

    MATH  Google Scholar 

Download references

Acknowledgement

The author should like to thank the referee for thoughtful comments, and identifying an error in the proof of Theorem 3.1.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

    Tom Sanders

Authors
  1. Tom Sanders
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tom Sanders.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sanders, T. Coset decision trees and the Fourier algebra. JAMA 144, 227–259 (2021). https://doi.org/10.1007/s11854-021-0179-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-021-0179-y

Search

Navigation