A note on Diophantine approximation (II)

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ 1 …, λ r be any real numbers. Then there exists a real number α such that for all integers u > 0 and for q = 1,…,r , where C = C(r) > 0. Thus all the numbers α+ λ 1 , …, α+ λ r are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ 1 , …, λ r and μ 1 , …, μ r there exist α, β such that for all integers u > 0 and for q=1,…, r , where again C = C(r) > 0 . Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.