Sum-product phenomena: \(\mathfrak{p}\)-adic case

Journal d'Analyse Mathématique volume 142, pages349–419(2020)Cite this article

Abstract

The sum-product phenomena over a finite extension K of ℚp is explored. The main feature of the results is the fact that the implied constants are independent of p.

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References

  1. [Bou03]

    J. Bourgain, On the Erdős–Volkmann and Katz–Tao ring conjectures, Geom. Funct. Anal. 13 (2003), 334–365.

    MathSciNet  Article  Google Scholar 

  2. [Bou05]

    J. Bourgain, Mordell’s exponential sum estimate revisited, J. Amer. Math. Soc. 18 (2005), 477–499.

    MathSciNet  Article  Google Scholar 

  3. [Bou08]

    J. Bourgain, The sum-product in ℤqwith q arbitrary, J. Anal. Math. 106 (2008), 1–93.

    MathSciNet  Article  Google Scholar 

  4. [BKT04]

    J. Bourgain, N. Katz and T. Tao, A sum-product estimate for finite fields and applications, Geom. Funct. Anal. 14 (2004), 27–57.

    MathSciNet  Article  Google Scholar 

  5. [BG09]

    J. Bourgain and A. Gamburd, Expansion and random walks in SLd(ℤ/pnℤ): II. With an appendix by J. Bourgain, J. Eur. Math. Soc. (JEMS) 11 (2009), 1057–1103.

    MathSciNet  Article  Google Scholar 

  6. [BG08]

    J. Bourgain and A. Gamburd, On the spectral gap for finitely-generated subgroups of SU(2), Invent. Math. 171 (2008), 83–121.

    MathSciNet  Article  Google Scholar 

  7. [BG12]

    J. Bourgain and A. Gamburd, A spectral gap theorem in SU(d), J. Eur. Math. Soc. (JEMS) 14 (2012) 1455–1511.

    MathSciNet  Article  Google Scholar 

  8. [BISG17]

    R. Boutonnet, A. Ioana and A. Salehi Golsefidy, Local spectral gap in simple Lie groups and application, Invent. Math. 208 (2017), 715–802.

    MathSciNet  Article  Google Scholar 

  9. [BGT11]

    E. Breuillard, B. Green and T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), 774–819.

    MathSciNet  Article  Google Scholar 

  10. [Cha02]

    M. Chang, A polynomial bound in Freiman’s theorem, Duke Math. J. 113 (2002), 399–419.

    MathSciNet  Article  Google Scholar 

  11. [CT06]

    T. Cover and J. Thomas, Elements of Information Theory, Wiley-Interscience, Hoboken, NJ, 2006.

    Google Scholar 

  12. [dS15]

    N. de Saxcé, A product theorem in simple Lie groups, Geom. Funct. Anal. 25 (2015), 915–941.

    MathSciNet  Article  Google Scholar 

  13. [EM03]

    G. Edgar and C. Miller, Borel subrings of the reals, Proc. Amer. Math. Soc. 131 (2003), 1121–1129.

    MathSciNet  Article  Google Scholar 

  14. [Gow08]

    T. Gowers, Quasirandom groups, Combin. Probab. Comput. 17 (2008), 363–387.

    MathSciNet  Article  Google Scholar 

  15. [Gow98]

    T. Gowers, A new proof of Szemeredi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551.

    MathSciNet  Article  Google Scholar 

  16. [Hel05]

    H. Helfgott, Growth and generation in SL2(ℤ/pℤ), Ann. of Math. (2) 167 (2008) 601–623.

    MathSciNet  Article  Google Scholar 

  17. [Hel11]

    H. Helfgott, Growth in SL3(ℤ/pℤ), J. Eur. Math. Soc. (JEMS) 13 (2011), 761–851.

    MathSciNet  Article  Google Scholar 

  18. [Joh92]

    A. Johnson, Measures on the circle invariant under multiplication by a nonlacunary sub-semigroup of the integers, Israel J. Math. 77 (1992), 211–240.

    MathSciNet  Article  Google Scholar 

  19. [KT01]

    N. Katz and T. Tao, Some connections between Falconer’s distance set conjecture, and sets of Furstenberg type, New York J. Math. 7 (2001) 149–187.

    MathSciNet  MATH  Google Scholar 

  20. [Lan94]

    S. Lang, Algebraic Number Theory, Springer, New York, 1994.

    Google Scholar 

  21. [LMP99]

    E. Lindenstrauss, D. Meiri and Y. Peres, Entropy of convolutions on the circle, Ann. of Math. (2) 149 (1999), 871–904.

    MathSciNet  Article  Google Scholar 

  22. [LV]

    E. Lindenstrauss and P. Varjú, Work in progress, June, 2014.

  23. [Man42]

    H. Mann, A Proof of the fundamental theorem on the density of sums of set of positive integers, Ann. of Math. (2) 43 (1942), 523–527.

    MathSciNet  Article  Google Scholar 

  24. [Neu99]

    J. Neukrich, Algebraic Number Theory; Springer-Verlag, Berlin, 1999.

    Google Scholar 

  25. [PS16]

    L. Pyber and E. Szabó, Growth in finite simple groups of Lie type, J. Amer. Math. Soc. 29 (2016) 95–146.

    MathSciNet  Article  Google Scholar 

  26. [Rud90]

    D. Rudolph, ×2 and ×3 invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), 395–406.

    MathSciNet  Article  Google Scholar 

  27. [SG17]

    A. Salehi Golsefidy, Super approximation, I: \(\mathfrak{p}\)-adic semisimple case, Int. Math. Res. Not. IMRN 2017 (2017), 7190–7263.

    MathSciNet  MATH  Google Scholar 

  28. [SG19]

    A. Salehi Golsefidy, Super-approximation, II: the p-adic case and the case of bounded powers of square-free integers, J. Eur. Math. Soc. (JEMS) 21 (2019), 2163–2232.

    MathSciNet  Article  Google Scholar 

  29. [SX91]

    P. Sarnak and X. Xue, Bounds for multiplicities of automorphic representations, Duke Math. J. 64 (1991), 207–227.

    MathSciNet  Article  Google Scholar 

  30. [Ser79]

    J.-P. Serre, Local Fields, Springer, New York, 1979.

    Google Scholar 

  31. [TV06]

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

    Google Scholar 

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Author information

Affiliations

  1. Mathematics Department, University of California, San Diego, CA, 92093-0112, USA

    Alireza Salehi Golsefidy

Authors
  1. Alireza Salehi Golsefidy
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Corresponding author

Correspondence to Alireza Salehi Golsefidy.

Additional information

A. S.-G. was partially supported by the NSF grant DMS-1303121, the A. P. Sloan Research Fellowship. Parts of this work was done when I was visiting Isaac Newton Institute and the MSRI, and I would like to thank both of these institutes for their hospitality.

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Golsefidy, A.S. Sum-product phenomena: \(\mathfrak{p}\)-adic case. JAMA 142, 349–419 (2020). https://doi.org/10.1007/s11854-020-0139-y

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