On the distribution of powers of a complex number |
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Authors: | Artūras Dubickas |
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Institution: | (3) Abt. Math. III, University Ulm, Ulm, Germany; |
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Abstract: | Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α n + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ
α
n
+ ν, n = 0, 1, 2, . . . , modulo
\mathbbZi],{\mathbb{Z}i],} where i=?{-1}{i=\sqrt{-1}} and
\mathbbZi]=\mathbbZ+i\mathbbZ{\mathbb{Z}i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any
z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo
\mathbbZi]{\mathbb{Z}i]}
the fractional part of z and write {z} for this, in general, complex number lying in the unit square
S:={z ? \mathbbC:0 £ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over
\mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ
α
n
+ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over
\mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤ 1 and
x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed
a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence
zn ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists
x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ
α
n
−z
n
, n = 0, 1, 2, . . . , all lie ‘far’ from the lattice
\mathbbZi]{\mathbb{Z}i]}. In particular, they all can be covered by a union of small discs with centers at
(1+i)/2+\mathbbZi]{(1+i)/2+\mathbb{Z}i]} if |α| is large. |
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Keywords: | |
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