Abstract
The goal of this chapter is to introduce and expound one of the most fundamental and useful theorems in Inverse Additive Theory: Kneser’s Theorem. As remarked before, we are interested in determining the structure of A and B with small sumset. We saw in the previous sections that taking A and B to both be arithmetic progressions of common difference generally gives |A+B|=|A|+|B|−1. However, if A=B=H is a subgroup, then A+B=2A=2H=H, and so |A+B|=|B|, which is considerably smaller. Kneser’s Theorem will tell us that filling up subgroups in A+B or, more generally, cosets (which are just translates of subgroups), is the only way to drop the cardinality of the sumset below the bound |A|+|B|−1.
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Grynkiewicz, D.J. (2013). Periodic Sets and Kneser’s Theorem. In: Structural Additive Theory. Developments in Mathematics, vol 30. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00416-7_6
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DOI: https://doi.org/10.1007/978-3-319-00416-7_6
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