اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIterated sumsets and Hilbert functions
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Let A be a finite subset of an abelian group (G, +). Let h $ge$ 2 be an integer. If |A| $ge$ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + $times$ $times$ $times$ + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It is known that |(h -- 1)A| $ge$ |hA| (h--1)/h , a consequence of Pl{u}nneckes inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h$ge$0 with the Hilbert function of a standard graded algebra. We then apply Macaulays 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| $ge$ $theta$(x, h) |hA| (h--1)/h for some factor $theta$(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that $theta$(x, h) asymptotically tends to e $approx$ 2.718 as |A| grows and h lies in a suitable range varying with |A|.
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