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Multiparameter Projection Theorems with Applications to Sums-Products and Finite Point Configurations in the Euclidean Setting

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Book cover Recent Advances in Harmonic Analysis and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 25))

Abstract

In this chapter we study multiparameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of \(A \cdot A + \cdots+ A \cdot A\), where A is a subset of the real line of a given Hausdorff dimension, \(A + A =\{ a + a^{\prime} : a,a^{\prime} \in A\}\) and A ⋅A={ a ⋅a′ :a,a′∈A}. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ℝ d is sufficiently large, then the \(({ k+1 \atop 2} )\)-dimensional Lebesgue measure of the set of k-simplexes determined by this set is positive. The sharpness of these results and connection with number theoretic estimates are also discussed.

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References

  1. Bourgain, J.: On the Erdős-Volkmann and Katz-Tao ring conjectures. GAFA, 13, 334–364 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourgain, J., Katz, N., Tao, T.: A sum–product estimate in finite fields, and applications. Geom. Func. Anal. 14, 27–57 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Edgar, G., Miller, C.: Borel sub-rings of the reals. PAMS, 131, 1121–1129 (2002)

    Article  MathSciNet  Google Scholar 

  4. Erdoğan, B.: A bilinear Fourier extension theorem and applications to the distance set problem, IMRN 23, 1411–1425 (2006)

    Google Scholar 

  5. Falconer, K.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  6. Falconer, K.: On the Hausdorff dimensions of distance sets. Mathematika 32, 206–212 (1986)

    Article  MathSciNet  Google Scholar 

  7. Hart, D., Iosevich, A.: Ubiquity of simplices in subsets of vectors spaces over finite fields. Anal. Mathematica 34, 29–38 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hart, D., Iosevich, A.: Sums and products in finite fields: An integral geometric viewpoint. Contemp. Math. Radon transforms, geometry, and wavelets 464, 129–135 (2008)

    Article  MathSciNet  Google Scholar 

  9. Hart, D., Iosevich, A., Alex, D.K., Rudnev, M.: Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture. Transactions of the AMS, 363, 3255–3275 (2011).

    Article  MATH  Google Scholar 

  10. Hart, D., Iosevich, A., Solymosi, J.: Sum-product estimates in finite fields via Kloosterman sums. Int. Math. Res. Not. IMRN 5, p. 4 (2007)

    MathSciNet  Google Scholar 

  11. Iosevich, A., Rudnev, M.: Erdös distance problem in vector spaces over finite fields. Trans. Amer. Math. Soc. 359(12), 6127–6142 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iosevich, A., Senger, S.: Sharpness of Falconer’s estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains (2010). (http://arxiv.org/pdf/1006.1397)

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

    MathSciNet  MATH  Google Scholar 

  14. Mattila, P.: On the Hausdorff dimension and capacities of intersections. Mathematika 32, 213–217 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mattila, P.: Geometry of Sets and Measures in Euclidean Space, vol. 44. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  16. Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions and the dimension of exceptions. Duke Math J. 102, 193–251 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Solomyak, B.: Measure and dimension of some fractal families. Math. Proc. Cambridge Phil. Soc. 124, 531–546 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Solymosi, J.: Bounding multiplicative energy by the sumset. Adv. Math. 222(2), 402–408 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tao, T., Vu, V.: Additive Combinatorics. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  20. Wolff, T.: Decay of circular means of Fourier transforms of measures. Int. Math. Res. Not. 10, 547–567 (1999)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana-Champaign, Illinois, USA

    Burk Erdoğan

  2. Department of Mathematics, Kansas State University, Manhattan, Kansas

    Derrick Hart

  3. Department of Mathematics, University of Rochester, Rochester, NY, USA

    Alex Iosevich

Authors
  1. Burk Erdoğan
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  2. Derrick Hart
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  3. Alex Iosevich
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Corresponding author

Correspondence to Alex Iosevich .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of South Carolina, Greene Street 1523, Columbia, 29208, South Carolina, USA

    Dmitriy Bilyk

  2. University Park, Department of Mathematics, Florida International University, SW 8th St., Rm 416 11200, Miami, 33199, Florida, USA

    Laura De Carli

  3. , Department of Mathematics, Universtity of Georgia, University of Georgia, Athens, 30602, Georgia, USA

    Alexander Petukhov

  4. Georgia Southern University, PO Box 8093, Statesboro, 30460, Georgia, USA

    Alexander M. Stokolos

  5. , Department of Mathematics, Georgia Institute of Technology, Cherry Street 686, Atlanta, 30332, Georgia, USA

    Brett D. Wick

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Erdoğan, B., Hart, D., Iosevich, A. (2012). Multiparameter Projection Theorems with Applications to Sums-Products and Finite Point Configurations in the Euclidean Setting. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_11

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