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Sums and k-sums in abelian groups of order k

✍ Scribed by Weidong Gao; Imre Leader

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
111 KB
Volume
120
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a 1 , . . . , a r that does not have 0 as a k-sum is attained at the sequence b 1 , . . . , b r-k+1 , 0, . . . , 0, where b 1 , . . . , b r-k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k -1 times. This proves a conjecture of BollobΓ‘s and Leader, and extends results of Gao and of BollobΓ‘s and Leader.

πŸ“œ SIMILAR VOLUMES

Sets in Abelian groups with distinct sum
Sets in Abelian groups with distinct sums of pairs
✍ Harri HaanpÀÀ; Patric R.J. Γ–stergΓ₯rd πŸ“‚ Article πŸ“… 2007 πŸ› Elsevier Science 🌐 English βš– 141 KB

A subset S = {s 1 , . . . , s k } of an Abelian group G is called an S t -set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S 2 -set in G. Let v(k) denote the order of the smallest Abelian group for which s(G) k. In this article, boun