Kevin O’Bryant, whose work on sets that have more sums than differences was mentioned in this recent post, writes as follows:

Here’s a related problem that Mel Nathanson and myself (with Ruzsa and a few students) have also been thinking about. For a finite set

Aof integers, letS= {a+b:a,b\inA} be the sumset, and letT= {2a+b:a,b\inA}. Can it happen thatSis larger thanT? The answer is yes (we have a proof) but we don’t have a single example. Our proof gives a set that would have at leaste^{2000}elements, so although our proof is “constructive”, it really isn’t in any practical sense. My professors thought “constructive” referred to a proof that didn’t use the axiom of choice, my generation thinks of it as an algorithm that runs in polynomial time, my students will have to worry about which polynomial. :)

Note that this problem, like the original MSTD problem, has the pleasant property of affine invariance: You can scale or translate the elements of the set by any constant, and the size of the *S* and *T* sets remains unchanged. In the case of an arithmetic progression, such as the *n*-element set {0, 1, 2,…, *n*–1}, the sumset *S* has 2*n*–1 elements, whereas the set *T* has 3*n*–2 elements. Thus |*T*| is greater than |*S*| by about *n*. For a random sampling of sets *other* than arithmetic progressions, it appears that |*T*| usually exceeds |*S*| by an even larger margin. I have no idea where to look for that elusive example with |*T*| < |*S*|.