On the possible orders of a basis for a finite cyclic group
The electronic journal of combinatorics, Tome 17 (2010)
We prove a result concerning the possible orders of a basis for the cyclic group ${\Bbb Z}_n$, namely: For each $k \in {\Bbb N}$ there exists a constant $c_k > 0$ such that, for all $n \in {\Bbb N}$, if $A \subseteq {\Bbb Z}_n$ is a basis of order greater than $n/k$, then the order of $A$ is within $c_k$ of $n/l$ for some integer $l \in [1,k]$. The proof makes use of various results in additive number theory concerning the growth of sumsets. Additionally, exact results are summarized for the possible basis orders greater than $n/4$ and less than $\sqrt{n}$. An equivalent problem in graph theory is discussed, with applications.
@article{10_37236_351,
author = {Peter Dukes and Peter Hegarty and Sarada Herke},
title = {On the possible orders of a basis for a finite cyclic group},
journal = {The electronic journal of combinatorics},
year = {2010},
volume = {17},
doi = {10.37236/351},
zbl = {1201.11017},
url = {http://geodesic.mathdoc.fr/articles/10.37236/351/}
}
Peter Dukes; Peter Hegarty; Sarada Herke. On the possible orders of a basis for a finite cyclic group. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/351
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