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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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[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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