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Weyl’s Equidistribution Theorem
N IKHIL A DDLEMAN
University of New Mexico
December 12, 2013
Introduction
Fourier Analysis has seen an incredible breadth of applications, ranging
from pure mathematics and physics, to innumerable applications in signalprocessing and computers. It is worth noting that in the development of the
Fourier theory and its extensions, a great number of powerful techniques
were produced incidentally. In this paper we will examine one such proof
technique which sees use in the proof of a remarkable result in number
theory: Weyl’s Equidistribution Theorem.
Terminology and Motivation
Equidistribution
A sequence of real numbers is said to be equidistributed if the quantity of
terms which fall within an interval is proportional only to the length of the
interval. Weyl’s Equidistribution theorem defines a class of such sequences:
the fractional parts of integer multiples of irrational numbers.
Equidistribution is a property of a number of sequences studied in Diophantine approximation theory. It is also a crucial property for Monte Carlo
integration, which is a method for the numerical calculation of integrals
using random numbers.
Notation
We must define our notation before proceeding. h x i = x mod 1 or in other
words, the fractional part of x. For a set S, card S denotes the set’s cardinality.
For our purposes, we will use T to denote the torus of length 2π
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The Theorem
If γ is irrational then for a, b ∈ [0, 1] we have
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card{1 ≤ r ≤ n : a ≤ hrγi ≤ b} → b − a as n → ∞
(1)
n
Note that for rational γ equidistribution will not occur. The fractional
parts of integer multiples of a rational number will always fall on multiples
of one over the denominator, so it is easy to find an interval in which a
multiple never falls.
Instead of proving this directly, we will prove two statements that imply
the desired result.
1 n
1
f (2πrγ) →
n r∑
2π
=1
Z
T
f (t)dt as n → ∞
1
card{1 ≤ r ≤ n : 2πrγ ∈ [2πa, 2πb]} → b − a as n → ∞
n
(1.1)
(1.2)
Define a function
1 n
1
Gn ( f ) = ∑ f (2πrγ) −
n r =1
2π
Z
T
f (t)dt
We will proceed by examining a sequence of simple, continuous functions.
Gn (1) =
1 n
1
1−
∑
n r =1
2π
Z
T
1dt = 1 − 1 = 0
Since our goal is proving Gn ( f ) → 0 for continuous functions, f , we will
next examine exponential functions and trigonometric polynomials, which
are dense in the continuous functions on T.
2
For eist with s ∈ Z, s 6= 0 and t ∈ T
Z
1 n
1
eist dt
Gn (eist ) = ∑ e2πirsγ −
n r =1
2π T
1
n
= e2πisγ ∑ e2πirsγ − 0
n
r =0
1
1 − e2πinsγ = e2πisγ
n
1 − e2πisγ 1 1 − e2πinsγ = n 1 − e2πisγ 1
2
≤
→ 0 as n → ∞
n |1 − e2πisγ |
Because s 6= 0 the right integral vanishes. We change the form of the
sum to use the formula for geometric series, and observe that the numerator
is always smaller than 2 while the denominator cannot equal 0.
Extending this result to trigonometric polynomials is easy as all of the
operations in G are linear.
ikt
For a trigonometric polynomial P = ∑m
k =−m ak e
m
Gn ( P) =
∑
ak Gn (eikt ) → 0
k=−m
as desired.
One of the most powerful and frequently used results is the density
of the trigonometric polynomials in the continuous functions on T. This
means for any f ∈ C(T) and e ≥ 0 there is a trigonometric polynomial g
with | f (t) − g(t)| ≤ e for all t ∈ T. And of course, if f andg are e − close on
T
| Gn ( f ) − Gn ( g)| ≤
1 n
1
| f (2πrγ) − g(2πrγ)| +
n r∑
2π
=1
Z 2π
0
| f (t) − g(t)|dt ≤ 2e
for all n.
Finally letting g be a trigonometric polynomial we will use the analysis
trick of adding and subtracting something we understand
| Gn ( f )| ≤ | Gn ( g)| + | Gn ( f ) − Gn ( g)| ≤ 2e
With this we have shown (1.1), and the beauty of Weyl’s method emerges.
This result relates a discrete sum with an integral, and approximates the
methods of Monte Carlo integration. Proving (1.2) is a simple matter of
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defining two continuous functions, f + and f − , to bound the cardinality
function over the desired interval, [2πa, 2πb].
f + (t) ≥ 1 ≥ f − (t) for all t ∈ [2πa, 2πb]
f + (t) ≥ 0 and f −R(t) = 0 for all t ∈
/ [2πa, 2πb]
1
(b −
a
)
+
e
≥
f
(
t
)
dt
+
2π
R
1
f
(
t
)
dt
≥
(
b
−
a) − e
−
2π
These functions can be used to count the instances of 2πrγ falling within
our interval.
n
∑
n
f + (2πrγ) ≥ card{1 ≤ r ≤ n : 2πrγ ∈ [2πa, 2πb]} ≥
r =1
∑ f − (2πrγ)
r =1
But allowing n → ∞ for any e ≥ 0 we have | Gn ( f + )| ≤ e and | Gn ( f − )| ≤ e.
1
2π
Z
T
f + (t)dt + e ≥
1
1
card{1 ≤ r ≤ n : 2πrγ ∈ [2πa, 2πb]} ≥
n
2π
Z
T
f + (t)dt + e
We defined our functions to give
(b − a) + 2e ≥
1
card{1 ≤ r ≤ n : 2πrγ ∈ [2πa, 2πb]} ≥ (b − a) − 2e
n
so, as desired
1
card{1 ≤ r ≤ n : 2πrγ ∈ [2πa, 2πb]} → (b − a) as n → ∞
n
Investigating other sequences
Weyl eventually proved a similar result for hr2 γi for r ∈ Z and opened
up an enduring question on the equidistribution of other sequences. One
which remains an open question is which γ have γr equidistibuted. We
do know, however, that it is at least not true for all irrational γ. Consider
Binet’s formula for the nth Fibonacci number
√ !n
√ !n
1+ 5
1− 5
Fn =
+
2
2
√ of course Fn is always an integer. Observe that −1 < 1−2 5 < 0 so
√ !n
1− 5
→ 0 as n → ∞
2
implying
so for γ =
√ !n
1+ 5
h
i → 1 as n → ∞
2
√
1+ 5
2
we have γn is not equidistributed.
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Some History
Hermann Weyl was a German mathematician who is also remembered for
his substantial contributions to theoretical physics, mathematical physics,
and philosophy. During his doctoral studies, Weyl worked under David
Hilbert. He successfully proved his equidistribution theorem in 1909, and
soon afterward he moved to Switzerland where he worked alongside Albert
Einstein and Erwin Schrodinger.
Weyl’s contributions to mathematics appear in a wide body of fields.
The Weyl Law and Weyl Conjecture remain prominent in spectral theory.
The prolific and substantial contributions from the man are patently obvious
when one notes the long list of topics in mathematics and physics named
after him. We have the Weyl group, in algebra, Weyl’s postulate on the
physics of gravity, and many others.
References
1. Korner, T.W. Fourier Analysis. Cambridge University Press. 1988.
2. O’Connor, John J.; Robertson, Edmund F. "Hermann Weyl" MacTutor
History of Mathematics archive 1973: Random House, N.Y.
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