Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — arf < c, r(q — fir) < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals a and /3, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46"" = 1/2.6043 • • •.

Let Pd denote the family of all polynomials of degree at most d in one variable x, with real coefficients. A sequence of positive numbers x1 ≤ x2 ≤ . . . is called Pd-controlling if there exist y1, y2, . . . ∈ R such that for every polynomial p ∈ Pd there exists an index i with |p(xi)−yi| ≤ 1. We settle an problem of Makai and Pach (1983) by showing that x1 ≤ x2 ≤ . . . is Pd-controlling if and...

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them togeth...