# A basis of ℤₘ, II

Colloquium Mathematicae (2007)

• Volume: 108, Issue: 1, page 141-145
• ISSN: 0010-1354

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## Abstract

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Given a set A ⊂ ℕ let ${\sigma }_{A}\left(n\right)$ 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, ${\sigma }_{A}\left(n\right)$ 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 ${\sigma }_{A}\left(n̅\right)\le 5120$ for all n̅ ∈ ℤₘ.

## How to cite

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Min Tang, and Yong-Gao Chen. "A basis of ℤₘ, II." Colloquium Mathematicae 108.1 (2007): 141-145. <http://eudml.org/doc/283413>.

@article{MinTang2007,
abstract = {Given a set A ⊂ ℕ let $σ_A(n)$ 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, $σ_A(n)$ 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 $σ_A(n̅) ≤ 5120$ for all n̅ ∈ ℤₘ.},
author = {Min Tang, Yong-Gao Chen},
journal = {Colloquium Mathematicae},
keywords = {Erdős–Turán conjecture; additive bases; representation function},
language = {eng},
number = {1},
pages = {141-145},
title = {A basis of ℤₘ, II},
url = {http://eudml.org/doc/283413},
volume = {108},
year = {2007},
}

TY - JOUR
AU - Min Tang
AU - Yong-Gao Chen
TI - A basis of ℤₘ, II
JO - Colloquium Mathematicae
PY - 2007
VL - 108
IS - 1
SP - 141
EP - 145
AB - Given a set A ⊂ ℕ let $σ_A(n)$ 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, $σ_A(n)$ 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 $σ_A(n̅) ≤ 5120$ for all n̅ ∈ ℤₘ.
LA - eng
KW - Erdős–Turán conjecture; additive bases; representation function
UR - http://eudml.org/doc/283413
ER -

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