Given a set A ⊂ ℕ let denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and for all n̅ ∈ ℤₘ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-12,
author = {Min Tang and Yong-Gao Chen},
title = {A basis of Zm, II},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {141-145},
zbl = {1187.11006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-12}
}
Min Tang; Yong-Gao Chen. A basis of ℤₘ, II. Colloquium Mathematicae, Tome 107 (2007) pp. 141-145. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-12/