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