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__Restricted Sums in Abelian Groups__

For two subsets *A* and *B* of an abelian group let *A+B={a+b}* be the set of all elements of the group, representable as a sum of an element from *A* and an element from *B*. Furthermore, let *A^B* be the subset of *A+B* consisting only of those elements representable as *a+b* with *a* and **b** *distinct*.

If there exists an element *c* in *A+B* with a unique representation as *c=a+b*, then the cardinality of *A+B* is at least *|A|+|B|-1*; this was proved by
P. Scherk in 1955.
I conjecture that under the same assumption one has a similar lower bound for the cardinality of *A^B*.

Conjecture.
If there exists an element *c* in *A+B* with a unique representation as *c=a+b*, then the cardinality of the restricted sum *A^B* is at least *|A|+|B|-3*.

An immediate consequence of this would be a new proof of the Erdos-Heilbronn conjecture, and there are other exiting conclusions.

Notice, that if *A=B* then the existence of an an element *c* with a unique representation implies that *A^B* is a *proper* subset of *A+B*, as this representation of *c* necessarily must be of the form *c=2a*. Moreover if, in addition to *A=B*, in the underlying group there are no elements of even order, then an element with a unique representation exists *if and only if* **A^B** is proper in *A+B*.

The method of Scherk is not particularly complicated, but depends heavily on the e-transform and thus seems to be inapplicable to restricted summation.