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On canonical expansions of integers in imaginary quadratic fields. (English)
Acta Math. Hung. 93, No.4, 347-357 (2001).
Let $ϑ$ be an algebraic integer. The pair $(θ, {\Cal A})$ with $θ\in {\bbfQ}[ϑ]$ and ${\Cal A} =\{0, \dots, |\text{Norm}(θ)|\}$ is called a number system in ${\bbfQ}[ϑ]$ if there exist for each $γ\in {\bbfQ}[ϑ]$ an $m \ge 0$ and $a_j \in {\Cal A}, j=0, \dots, m$ such that $γ= \sum_{j=0}^m a_j θ^j$. {\it I. Kátai} and {\it B. Kovács} [Acta Sci. Math. 42, 99-107 (1980; Zbl 0426.10011), Acta Math. Acad. Sci. Hung. 37, 159-164 (1981; Zbl 0477.10012)] and independently {\it W. J. Gilbert} [J. Math. Anal. Appl. 83, 264-274 (1981; Zbl 0472.10011)] proved that if the minimal polynomial of $θ$ is $x^2 + Ex + F$ then $(θ, {\Cal A})$ is a number system if and only if $F \ge 2$ and $-1 \le E \le F$. An element $π\in {\bbfQ}[θ]$ is called periodic if there is an $l > 0$ such that $π= \sum_{j=0}^{l-1} a_j θ^j + πθ^l$. The aim of the present paper is to describe the set of periodic elements $\Cal P$ and the possible periods in the ring of integers of imaginary quadratic fields. Let ${\bbfQ}(θ)={\bbfQ}(i\sqrt{D})$ with a square-free integer $D>0$. Then $θ= a + b ω$, where $ω=i\sqrt{D}$ or $(1+ i\sqrt{D})/2$ according as $D \equiv 1,2 \pmod{4}$ or $D \equiv 3 \pmod{4}$. The size of $\Cal P$ depends on these cases. Assume $b\ge 1$. If $D \equiv 1,2 \pmod{4}$ and $a \ge 1$ then $\# {\Cal P} = b+1$, while if $a\le -1$ then $\# {\Cal P} = b$. If $D \equiv 3 \pmod{4}$ then $\# {\Cal P}$ can take the values $b, b+1$ and $b+2$ only depending on $a,b$ and $D$.
Reviewer: A.Pethő (Debrecen)