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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
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