Additive Number Theory pp 195-202 | Cite as

# The Erdős–Turán Problem in Infinite Groups

## Summary

Let *G* be an infinite abelian group with | 2*G* |=| *G* |. We show that if *G* is not the direct sum of a group of exponent 3 and the group of order 2, then *G* possesses a perfect additive basis; that is, there is a subset *S* ⊆ *G* such that every element of *G* is uniquely representable as a sum of two elements of *S*. Moreover, if *Gis* the direct sum of a group of exponent 3 and the group of order 2, then it does not have a perfect additive basis; however, in this case, there exists a basis *S* ⊆ *G* such that every element of *G* has at most two representations (distinct under permuting the summands) as a sum of two elements of *S*. This solves completely the Erdős–Turán problem for infinite groups. It is also shown that if *G* is an abelian group of exponent 2, then there is a subset *S* ⊆ *G* such that every element of *G* has a representation as a sum of two elements of *S*, and the number of representations of nonzero elements is bounded by an absolute constant.

## Keywords

Additive basis Erdős–Turán problem Representation function## References

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