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Abstract

We prove results on the structure of a subset of the circle group having positive inner Haar measure and doubling constant close to the minimum. These results go towards a continuous analogue in the circle of Freiman’s 3k−4 theorem from the integer setting. An analogue of this theorem in Zp has been pursued extensively, and we use some recent results in this direction. For instance, obtaining a continuous analogue of a result of Serra and Zémor, we prove that if a subset A of the circle is not too large and has doubling constant at most 2+ε with ε< 10−4⁠, then for some integer n> 0⁠, the dilate n⋅A is included in an interval in which it has density at least 1/(1+ε)⁠. Our arguments yield other variants of this result as well, notably a version for two sets which makes progress towards a conjecture of Bilu. We include two applications of these results. The first is a new upper bound on the size of k-sum-free sets in the circle and in Zp⁠. The second gives structural information on subsets of ℝ of doubling constant at most 3+ε⁠.

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