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On linear versions of some addition theorems. (English) Zbl 1263.11015
Summary: Let $$K \subset L$$ be a field extension. Given $$K$$-subspaces $$A, B$$ of $$L$$, we study the subspace $$\langle AB\rangle$$ spanned by the product set $$AB = \{ab\mid a \in A,\, b \in B\}$$. We obtain some lower bounds on $$\dim_K\langle AB\rangle$$ and $$\dim_K\langle B^n\rangle$$ in terms of $$\dim_K A$$, $$\dim_K B$$ and $$n$$. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.

MSC:
 11B30 Arithmetic combinatorics; higher degree uniformity 11B75 Other combinatorial number theory 15A04 Linear transformations, semilinear transformations
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References:
 [1] DOI: 10.1006/jnth.2002.2793 · Zbl 1034.11020 · doi:10.1006/jnth.2002.2793 [2] Kemperman JHB, Indag. Math. 18 pp 247– (1956) · doi:10.1016/S1385-7258(56)50032-7 [3] DOI: 10.1007/BF01174162 · Zbl 0051.28104 · doi:10.1007/BF01174162 [4] DOI: 10.1007/BF01181357 · Zbl 0064.04305 · doi:10.1007/BF01181357 [5] DOI: 10.1016/0022-314X(84)90047-7 · Zbl 0524.10043 · doi:10.1016/0022-314X(84)90047-7
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