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On inhomogeneous diophantine approximation. (English) Zbl 0820.11042
Let $$\alpha$$ and $$\beta$$ be irrational numbers, $$\beta$$ not of the form $$m\alpha+ n$$ ($$m$$, $$n$$ integers) and define $M(\alpha, \beta)= \liminf \{| q|\;\| q\alpha- \beta\|:\;| q|\to \infty\}$ to be the inhomogeneous approximation constant for the pair $$\alpha$$, $$\beta$$. Here $$\| x\|$$ means the distance of $$x$$ from the nearest integer. Further let $$M(\alpha)= \sup M(\alpha, \beta)$$. It follows from an old result by Khintchine that $$M(\alpha)< 1/4$$ holds when the simple continued fraction expansion for $$\alpha$$ has bounded partial quotients. In particular, for $$\alpha_ 0= (1+ \sqrt {5}) /2$$ one has $$M(\alpha_ 0)= 1/(4 \sqrt{5})= \delta_ 0$$. The authors give a new proof for this.
Moreover, they find an infinite sequence of successive minima $$\delta_ 0> \delta_ 1> \delta_ 2> \dots$$ with $$\lim \delta_ n= (2(5+ \sqrt {5}) )^{-1}$$ such that, for each $$n\geq 0$$, $$M(\alpha_ 0, \beta)= \delta_ n$$ is true precisely when $$\beta$$ belongs to a certain specified set $$S_ n$$. For each $$n\geq 0$$, if $$\beta$$ does not belong to any of the sets $$S_ k$$, $$0\leq k\leq n$$, then $$M(\alpha_ 0, \beta)\leq \delta_{n+1}$$. For this purpose, the authors use a result by V. Sós to introduce a certain infinite series representation of a real number $$\beta$$, depending on the c.f. expansion of $$\alpha$$. They also use this expansion to study the restricted inhomogeneous constants $M_ + (\alpha, \beta)= \liminf \{q\| q\alpha- \beta \|:\;q\to \infty\} \;\text{ and } \;M_ - (\alpha, \beta)= \liminf \{-q \| q\alpha- \beta \|:\;q\to -\infty\}$ and to determine the approximation constants for certain pairs $$(\alpha_ 0, \beta)$$.
Reviewer: G.Ramharter (Wien)

##### MSC:
 11J20 Inhomogeneous linear forms 11H50 Minima of forms
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