Exponential sum and the equidistribution theorem

Consider the sequence , where gives the fractional part of a real number. The equidistribution theorem states that is irrational iff for any sub-interval of the unit interval, for sufficiently large , roughly of the numbers fall inside . More precisely, (Equidistributed sequence) A sequence with is equidistributed mod 1 if for every ,     (Equidistribution theorem) The sequence is equidistributed mod 1 iff is irrational. Let us try to understand why the sequence behaves differently depending on whether is rational or not. If for some integers and , then roughly of the first multiples of are integers, i.e., . In fact, it is not difficult to see that for , […]