Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-5,
author = {Xingwu Xia and Yongke Qu and Guoyou Qian},
title = {A characterization of sequences with the minimum number of k-sums modulo k},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {51-56},
zbl = {1301.11011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-5}
}
Xingwu Xia; Yongke Qu; Guoyou Qian. A characterization of sequences with the minimum number of k-sums modulo k. Colloquium Mathematicae, Tome 135 (2014) pp. 51-56. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-5/