Sumsets and Veronese varieties

In this paper, to any subset A ⊂ Z n we explicitly associate a unique monomial projection Y n , d A of a Veronese variety, whose Hilbert function coincides with the cardinality of the t -fold sumsets t A . This link allows us to tackle the classical problem of determining the polynomial p A ∈ Q [ t ] such that | t A | = p A ( t ) for all t ≥ t 0 and the minimum integer n 0 ( A ) ≤ t 0 for which this condition is satisfied, i.e. the so-called phase transition of | t A | . We use the Castelnuovo–Mumford regularity and the geometry of Y n , d A to describe the polynomial p A ( t ) and to derive new bounds for n 0 ( A ) under some technical assumptions on the convex hull of A ; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Y n , d A .