- 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. 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 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 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**wrote:*<iosevich@...>*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 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 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