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Let α and β be two irrational real numbers satisfying α ± β ∉ Z . We prove several inequalities between min k ∈ { 1 , … , n } ⁡ ‖ k α ‖ and min k ∈ S ⁡ ‖ k β ‖ , 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 min k ∈ { 1 , … , n } ⁡ ‖ k α ‖ and min k ∈ { 1 , … , n } ⁡ ‖ k β ‖ changes its sign infinitely often) and its variations are best possible. Some of the results are given in terms of the sequence d ( n ) = d α , 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 . [ABSTRACT FROM AUTHOR]