# A basis of Zₘ

Colloquium Mathematicae (2006)

• Volume: 104, Issue: 1, page 99-103
• ISSN: 0010-1354

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## Abstract

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Let ${\sigma }_{A}\left(n\right)=|\left(a,{a}^{\text{'}}\right)\in A²:a+{a}^{\text{'}}=n|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, ${\sigma }_{A}\left(n\right)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which ${\sigma }_{A}\left(n\right)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and ${\sigma }_{A}\left(n̅\right)\le 768$ for all n̅ ∈ Zₘ.

## How to cite

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Min Tang, and Yong-Gao Chen. "A basis of Zₘ." Colloquium Mathematicae 104.1 (2006): 99-103. <http://eudml.org/doc/286391>.

@article{MinTang2006,
abstract = {Let $σ_\{A\}(n) = |\{(a,a^\{\prime \}) ∈ A²: a + a^\{\prime \} = n\}|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_\{A\}(n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_\{A\}(n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_\{A\}(n̅) ≤ 768$ for all n̅ ∈ Zₘ.},
author = {Min Tang, Yong-Gao Chen},
journal = {Colloquium Mathematicae},
keywords = {Erdős-Turán conjecture; basis; congruence},
language = {eng},
number = {1},
pages = {99-103},
title = {A basis of Zₘ},
url = {http://eudml.org/doc/286391},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Min Tang
AU - Yong-Gao Chen
TI - A basis of Zₘ
JO - Colloquium Mathematicae
PY - 2006
VL - 104
IS - 1
SP - 99
EP - 103
AB - Let $σ_{A}(n) = |{(a,a^{\prime }) ∈ A²: a + a^{\prime } = n}|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A}(n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A}(n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A}(n̅) ≤ 768$ for all n̅ ∈ Zₘ.
LA - eng
KW - Erdős-Turán conjecture; basis; congruence
UR - http://eudml.org/doc/286391
ER -

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