## Bourgain's Proof of Katznelson Furstenberg weiss theorem

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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
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
• 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
• 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 youSincerelyDebashish --- On Sat, 7/12/08, Alex Iosevich wrote:From: Alex Iosevich Subject: Re: [harmonic] Bourgain's Proof of Katznelson Furstenberg weiss theoremTo: harmonicanalysis@yahoogroups.comDate: Saturday, July 12, 2008, 4:32 AMDear 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 IosevichDepartment of MathematicsUniversity of Missouri-ColumbiaColumbia, Missouri, 65211 USAhttp://www.math. missouri. edu/~iosevich On 7/11/08, noone wrote:While reading the book Geometric Discrepancy I came across the proofby Bourgain of Furstenberg, Katznelson and Weiss Theorem regarding theun-avoidability of all large distances in a set with positiveasymptotic density. Which I managed to follow ... Having gone through that I got hold of the original paper by Bourgainwhere 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 canfind more details. Thanking youDebashish-- I am constantly amazed by man's inhumanity to man Primo Levi

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