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