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Zbl 0745.11010
Puš, Vladimír
On multiplicative bases in Abelian groups. (English)
[J] Czech. Math. J. 41(116), No.2, 282-287 (1991). ISSN 0011-4642

Suppose that $M$ is a subset of a commutative semigroup $X$, and $k\geq 2$ an integer. Let $f\sb{M,k}(x)$ denote the number of essentially different solutions of $x=m\sb 1\dots m\sb k$ with $m\sb i\in M$ for $i=1,\dots,k$. An old conjecture of Erd\H{o}s and Turán asserts that, in the semigroup of positive integers with addition of integers, $f\sb{M,2}(n)>0$ for large $n$ implies that $\limsup\sb{n\to\infty}f\sb{M,2}(n)=\infty$. The conjecture is still open. The author proves the following theorem. Let $k\geq 2$ be any integer. Suppose that $G$ is an infinite abelian group such that (1) the cardinality $\alpha$ of $G$ is regular, and (2) for every $a\in G$ and every integer $r\in[2,2k]$, the number of the solutions of $x\sp r=a$ is less than $\alpha$. Then for every function $f: G\to\alpha\backslash\{0\}$ there exists a subset $M$ of $G$ such that $f\sb{M,k}(x)=f(x)$.
[X.-D.Jia (San Marcos)]

MSC 2000:: *11B13 Additive bases
20K01 Finite abelian groups

Keywords: multiplicative bases; representation function; infinite abelian group

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