On rational approximations to two irrational numbers

Let α and β be two irrational real numbers satisfying α±β∉Z. We prove several inequalities between mink∈{1,…,n}⁡‖kα‖ and mink∈S⁡‖kβ‖, where S is a set of positive integers, e.g., S={n}, S={1,…,n−1} or S={1,…,n} and ‖x‖ stands for the distance between x∈R and the nearest integer. We also give some constructions of α and β which show that the result of Kan and Moshchevitin (asserting that the difference between mink∈{1,…,n}⁡‖kα‖ and mink∈{1,…,n}⁡‖kβ‖ changes its sign infinitely often) and its variations are best possible. Some of the results are given in terms of the sequence d(n)=dα,β(n) defined as the difference between reciprocals of these two quantities. In particular, we prove that the sequence d(n) is unbounded for any irrational α,β satisfying α±β∉Z.