Positive projections
Update: 2011-07-07
Description
If A is a set of n positive integers, how small can the set { (a,b) : a,b in A } be? Here as usual (a,b) denotes the HCF of a and b. This elegant question was raised by Granville and Roesler, who also reformulated it in the following way: given a set A of n points in the integer grid Z^d, how small can (A-A)^+, the projection of the difference set of A onto the positive orthant, be? Freiman and Lev gave an example to show that (in any dimension) the size can be as small as n^{2/3} (up to a constant factor). Granville and Roesler proved that in two dimensions this bound is correct, i.e. that the size is always at least n^{2/3}, and they asked if this holds in any dimension. After some background material, the talk will focus on recent developments, including a negative answer to the n^{2/3} question. (joint work with Bela Bollobas)
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