Equidistribution mod 1

In mathematics, a sequence { a_n : n = 1, 2, 3, ... } is equidistributed modulo 1 precisely if for every interval (a, b) within the larger interval [0, 1),

<math>\lim_{n\to\infty}{\left|\left\{\,k\in\{\,1,\dots,n\,\} : a_k-\lfloor a_k \rfloor\in (a,b) \,\right\}\right| \over n}=b-a.<math>

In other words, the long-run proportion of fractional parts of a_n that fall within any subinterval is just the length of the subinterval. For example, since the interval (0.5, 0.8) occupies 30% of the space within the larger interval (0, 1), the proportion of members of the sequence whose fractional part falls between 0.5 and 0.8 will approach 0.3.

It can be shown that for any irrational number a, the sequence

a, 2a, 3a, ...

is equidistributed mod 1. A powerful general result is Weyl's criterion, which shows that equidistribution is equivalent to having a non-trivial estimate for the exponential sums formed with the sequence as exponents. For example it reduces this case of the multiples of a to summing finite geometric series.