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Bourgain's Proof of Katznelson Furstenberg weiss theorem

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  • noone
    While reading the book Geometric Discrepancy I came across the proof by Bourgain of Furstenberg,Katznelson and Weiss Theorem regarding the un-avoidability of
    Message 1 of 3 , Jul 11, 2008
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      While reading the book Geometric Discrepancy I came across the proof
      by Bourgain of Furstenberg,Katznelson and Weiss Theorem regarding the
      un-avoidability of all large distances in a set with positive
      asymptotic density. Which I managed to follow ...

      Having gone through that I got hold of the original paper by Bourgain
      where there is stronger version claiming the existence of some r,
      such that given any s>r there is a x \in A s.t \forall t \in [r,s]
      there exist y \in A with |x-y|= t .

      I am unable to figure out how the proof works ...

      Can you kindly give me a hint or direct me to a reference where I can
      find more details.

      Thanking you
      Debashish
    • Alex Iosevich
      Dear Debashish, A very clear and complete proof of what you are asking about can be found in a paper by Mihalis Kolountzakis ( Kolountzakis, M.
      Message 2 of 3 , Jul 12, 2008
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        Dear Debashish,
         
        A very clear and complete proof of what you are asking about can be found in a paper by Mihalis Kolountzakis ( Kolountzakis, M. N. Distance sets corresponding to convex bodies. Geom. Funct. Anal. 14 (2004), no. 4, 734--744). You can also get this paper directly from Mihalis' web page at the University of Crete). For a finite geometric version, you can take a look at  Iosevich, A.; Rudnev, M. Erdös distance problem in vector spaces over finite fields. Trans. Amer. Math. Soc. 359 (2007), no. 12, 6127--6142 and Hart, Derrick; Iosevich, Alex Ubiquity of simplices in subsets of vector spaces over finite fields. Anal. Math. 34 (2008), no. 1, 29--38.
         
        Best regards,
         
        Alex Iosevich
        Department of Mathematics
        University of Missouri-Columbia
        Columbia, Missouri, 65211 USA


         
        On 7/11/08, noone <d_bose2000@...> wrote:

        While reading the book Geometric Discrepancy I came across the proof
        by Bourgain of Furstenberg,Katznelson and Weiss Theorem regarding the
        un-avoidability of all large distances in a set with positive
        asymptotic density. Which I managed to follow ...

        Having gone through that I got hold of the original paper by Bourgain
        where there is stronger version claiming the existence of some r,
        such that given any s>r there is a x \in A s.t \forall t \in [r,s]
        there exist y \in A with |x-y|= t .

        I am unable to figure out how the proof works ...

        Can you kindly give me a hint or direct me to a reference where I can
        find more details.

        Thanking you
        Debashish




        --
        I am constantly amazed by man's inhumanity to man

        Primo Levi
      • bose D
        Dear Sir, Thanks for your kind reply and references. Indeed the paper you mentioned (Distance sets corresponding to convex bodies ) contains a detailed step by
        Message 3 of 3 , Jul 12, 2008
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          Dear Sir,

          Thanks for your kind reply and references.

          Indeed the paper you mentioned (Distance sets corresponding to convex bodies ) contains a detailed step by step proof of FKW theorem by Bourgain.

          However my question was regarding the farther generalization of this theorem in 2-dimension  in Bourgain's paper. My question was regarding that.

          Specifically
           
          After proving the FKW theorem he writes (Page 309 last para) as a remark:-

          \begin{remark}
          Combined with the results on the spherical maximal function in the plane, Theorem 1 can be improved as follows:
          \end{remark}

          \begin{theorem}"\

          If $A \subset \R^2$, $\delta(A) >0 $, there exists $l = l(A)$ such that whenever $l_1 > l$ there is a point $x \in A$
          fulfilling the condition $\{ |x-y|; y\in A \} \supset [l,l_1] $

          \end{theorem}

          my difficulty is with this part.

          I would be happy if you could explain the outline of the proof.

          Thanking you

          Sincerely
          Debashish



          --- On Sat, 7/12/08, Alex Iosevich <iosevich@...> wrote:
          From: Alex Iosevich <iosevich@...>
          Subject: Re: [harmonic] Bourgain's Proof of Katznelson Furstenberg weiss theorem
          To: harmonicanalysis@yahoogroups.com
          Date: Saturday, July 12, 2008, 4:32 AM

          Dear Debashish,
           
          A very clear and complete proof of what you are asking about can be found in a paper by Mihalis Kolountzakis ( Kolountzakis, M. N. Distance sets corresponding to convex bodies. Geom. Funct. Anal. 14 (2004), no. 4, 734--744). You can also get this paper directly from Mihalis' web page at the University of Crete). For a finite geometric version, you can take a look at  Iosevich, A.; Rudnev, M. Erdös distance problem in vector spaces over finite fields. Trans. Amer. Math. Soc.. 359 (2007), no. 12, 6127--6142 and Hart, Derrick; Iosevich, Alex Ubiquity of simplices in subsets of vector spaces over finite fields. Anal. Math. 34 (2008), no. 1, 29--38.
           
          Best regards,
           
          Alex Iosevich
          Department of Mathematics
          University of Missouri-Columbia
          Columbia, Missouri, 65211 USA


           
          On 7/11/08, noone <d_bose2000@yahoo. com> wrote:

          While reading the book Geometric Discrepancy I came across the proof
          by Bourgain of Furstenberg, Katznelson and Weiss Theorem regarding the
          un-avoidability of all large distances in a set with positive
          asymptotic density. Which I managed to follow ...

          Having gone through that I got hold of the original paper by Bourgain
          where there is stronger version claiming the existence of some r,
          such that given any s>r there is a x \in A s.t \forall t \in [r,s]
          there exist y \in A with |x-y|= t .

          I am unable to figure out how the proof works ...

          Can you kindly give me a hint or direct me to a reference where I can
          find more details.

          Thanking you
          Debashish




          --
          I am constantly amazed by man's inhumanity to man

          Primo Levi

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