On inhomogeneous diophantine approximation
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 \((\alpha_ 0, \beta)\).
scientific article; zbMATH DE number 645299