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Small doubling in a free group

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OOEW04 - Structure and Randomness - a celebration of the mathematics of Timothy Gowers

Let $A$ be a finite set in a free group, $|A| =n$. If $|A+A| = o(n)$, thenall but $o(n)$ elements of $A$ lie in a cyclic subgroup. The exponent 3/2is best possible. Unlike the commutative case, no such structural result follows from an estimateon the difference set $A-A$. However, if both difference sets $A-A$ and $-A+A$have size $O(n{9/8})$, then a similar conclusion follows. This exponent is probablynot optimal.

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