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,





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
,
fraction of the sequence equals
. So equidistribution does not occur when
is rational.
Now for , consider the exponential function
, which has period
. As discussed above, roughly
of the sequence
is
, so the exponential sum
contains an infinite number of ones. For example if
, the exponential sum evaluates to
, which is Grandi’s series and diverges. For an arbitrary
, the exponential sum is a repeating sum of all
-th roots of unity, and one can use the same technique showing that Grandi’s series diverges to establish that this exponential sum diverges too. However, the case when
is irrational differs: every nonzero integer multiple of
is an non-integer, and thus
for any
. Writing
, the exponential sum
converges to the geometric series
, since
.
In fact, the above intuition can be summarized into the following criterion due to Weyl:


From the above discussion, Weyl’s criterion immediately implies the equidistribution theorem.











Now suppose is irrational. Since
, we have
which tends to as
.
Before proving Weyl’s criterion, it is instructive to state the notion of equidistribution analytically. For , let
iff
. Note that
is a periodic function on all of
.


1) For

2) For all integers ,
Written in this form, Weyl’s criterion is asserting that if the sum of every interval function at a sequence converges to its integral, then every exponential sum converges to its integral as well, and vice versa. The reason is that both the space of interval functions and the space of exponential functions can approximate one another. More precisely, an exponential function is clearly periodic and continuous (and thus uniformly continuous), and every uniformly continuous function can be approximated by a finite sum of interval functions. For the other direction, a periodic, interval function can be approximated by a periodic, continuous function, which by Fejér’s Theorem can also be approximated by a finite sum of exponential functions. Now we have all the machinery to write down a proof.



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By linearity, we have
By taking sufficiently large and repeatedly applying triangle inequality, we have
implying that converges to
as
.
Now we consider the reverse direction (2) => (1). Fix and let
. Define
Note that is continuous, periodic, and dominates
. By Fejér’s Theorem, there exists
such that
. By linearity, we have
By taking sufficiently large and repeatedly applying triangle inequality, we have
implying that
Since dominates
, we have
One can similarly define a periodic, continuous function that approximates
from below, showing that
is bounded below by
, which together with the proceeding argument will show that the limit of
must exist and converge to
.