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A generalization of an addition theorem of Kneser. (English) Zbl 1034.11020
The authors prove the following unusual generalization of Kneser’s theorem on the cardinality of sumsets in commutative groups. Let $$E\subset K$$ be fields, $$A,B$$ finite dimensional subspaces of $$K$$ over $$E$$. Assume that every algebraic element in $$K$$ is separable over $$E$$. Then $\dim AB \geq \dim A + \dim B - \dim H,$ where $$\dim$$ means dimension over $$E$$ and $$H= \{x\in K: xAB\subset AB \}$$.
The connection with Kneser’s theorem is not obvious; the authors use a new (at least to the reviewer) method to embed a group into a field so that cardinalities of subsets are transformed into dimensions of subspaces.
It is not yet clear whether this approach leads to new advances in additive number theory, but it presents a refreshing new look at some classical methods and results.

MSC:
 11B75 Other combinatorial number theory 11B13 Additive bases, including sumsets
sumsets
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References:
 [1] Cassels, J.W.S., Local fields, (1986), Cambridge Univ. Press Cambridge, New York · Zbl 0595.12006 [2] Hou, X.D.; Leung, K.H.; Xiang, Q., New partial difference sets in Zp2t and a related problem about Galois rings, Finite fields appl., 7, 165-188, (2001) · Zbl 1001.11052 [3] Kneser, M., Abschätzung der asymptotischen dichte von summenmengen, Math. zeit., 58, 459-484, (1953) · Zbl 0051.28104 [4] Mann, H.B., Addition theorems, (1965), Wiley New York · Zbl 0189.29701 [5] Nathanson, M.B., Additive number theory, inverse problems and the geometry of sumsets, (1996), Springer New York · Zbl 0859.11003
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