Abstract
Let α1, …,αr ∈ R be ‘not very well approximable’, for example, Q-linearly independent real algebraic numbers. Then there are infinitely many positive square-free integers n such that ‖nαi‖ << n−(2/3r)+ϵ(1 ≤ i ≤ r), where ‖·‖ denotes distance to the nearest integer.
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