Let $\sigma_A(n)=|\{(a,a')\in A^2: a+a'=n\}|$,
where $n\in {\bf N}$ and $A$ is a subset of ${\bf N}$. Erdős
and Turán conjectured that for any basis $A$ of order 2 of
${\bf N}$, $\sigma_A(n)$ is unbounded. In 1990, Imre Z. Ruzsa
constructed a basis $A$ of order 2 of ${\bf N}$ for which $\sigma_A(n)$
is bounded in the square mean. In this paper, we show that there
exists a positive integer $m_0$ such that, for any integer $m\geq
m_0$, we have a set $A\subset {\bf Z}_m$ such that
$A+A={\bf Z}_m$ and $\sigma_A(\overline{n})\leq 768$ for all
$\overline{n}\in {\bf Z}_m$.
Mots-clés :
sigma where subset erd tur conjectured basis order sigma unbounded imre nbsp ruzsa constructed basis order nbsp which sigma bounded square mean paper there exists positive integer integer geq have set subset sigma overline leq overline
Affiliations des auteurs :
Min Tang
1
;
Yong-Gao Chen
2
1
Department of Mathematics Anhui Normal University Wuhu 241000, China
2
Department of Mathematics Nanjing Normal University Nanjing 210097, China
@article{10_4064_cm104_1_6,
author = {Min Tang and Yong-Gao Chen},
title = {A basis of ${\bf Z}_m$},
journal = {Colloquium Mathematicum},
pages = {99--103},
year = {2006},
volume = {104},
number = {1},
doi = {10.4064/cm104-1-6},
language = {de},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm104-1-6/}
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AU - Yong-Gao Chen
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Min Tang; Yong-Gao Chen. A basis of ${\bf Z}_m$. Colloquium Mathematicum, Tome 104 (2006) no. 1, pp. 99-103. doi: 10.4064/cm104-1-6
