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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. 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.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  23. 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.

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. 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. 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.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

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Authors and Affiliations

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

    Alireza Salehi Golsefidy

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  1. Alireza Salehi Golsefidy
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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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  • DOI: https://doi.org/10.1007/s11854-020-0139-y

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