# A basis of Zₘ

Colloquium Mathematicae (2006)

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

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topMin 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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