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An analogue of Vosper’s theorem for extension fields. (English) Zbl 1405.11134
Summary: We are interested in characterising pairs $$S, T$$ of $$F$$-linear subspaces in a field extension $$L/F$$ such that the linear span $$ST$$ of the set of products of elements of $$S$$ and of elements of $$T$$ has small dimension. Our central result is a linear analogue of Vosper’s theorem, which gives the structure of vector spaces $$S, T$$ in a prime extension $$L$$ of a finite field $$F$$ for which $\dim_FST =\dim_F S+\dim_F T-1,$ when $$\dim_FS, \dim_FT\geq 2$$ and $$\dim_FST\leq [L : F]-2$$.

##### MSC:
 11P70 Inverse problems of additive number theory, including sumsets 05E30 Association schemes, strongly regular graphs 11B30 Arithmetic combinatorics; higher degree uniformity 12F10 Separable extensions, Galois theory
##### Keywords:
Vosper’s theorem; $$F$$-linear subspaces; field extension
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