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Towards a function field version of Freiman’s theorem. (English) Zbl 1439.11283
Summary: We discuss a multiplicative counterpart of Freiman’s $$3k-4$$ theorem in the context of a function field $$F$$ over an algebraically closed field $$K$$. Such a theorem would give a precise description of subspaces $$S$$, such that the space $$S^2$$ spanned by products of elements of $$S$$ satisfies $$\mathrm{dim} S^2\leq 3 \mathrm{dim} S-4$$. We make a step in this direction by giving a complete characterisation of spaces $$S$$ such that $$\mathrm{dim} S^2=2 \mathrm{dim} S$$. We show that, up to multiplication by a constant field element, such a space $$S$$ is included in a function field of genus $$0$$ or $$1$$. In particular if the genus is $$1$$ then this space is a Riemann-Roch space.
##### MSC:
 11R58 Arithmetic theory of algebraic function fields 11P99 Additive number theory; partitions 05E40 Combinatorial aspects of commutative algebra 14H05 Algebraic functions and function fields in algebraic geometry
##### Keywords:
additive combinatorics; function fields
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##### References:
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