Tiểu Luận A quantitative version of the idempotent theorem in harmonic analysis

A quantitative version of the
idempotent theorem in harmonic analysis
By Ben Green* and Tom Sanders
Abstract
Suppose that G is a locally compact abelian group, and write M(G) for
the algebra of bounded, regular, complex-valued measures under convolution.
A measure  2 M(G) is said to be idempotent if    = , or alternatively if b 
takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem
states that a measure  is idempotent if and only if the set f
2
b
G : b (
) = 1g
belongs to the coset ring of
b
G, that is to say we may write
b  =
L X
j =1
1
j + j
where the j are open subgroups of
b
G.
In this paper we show that L can be bounded in terms of the norm kk,
and in fact one may take L 6 exp exp(Ckk
4
). In particular our result is
nontrivial even for nite groups.
1. Introduction
Let us begin by stating the idempotent theorem. Let G be a locally
compact abelian group with dual group
b
G. Let M(G) denote the measure
algebra of G, that is to say the algebra of bounded, regular, complex-valued
measures on G. We will not dwell on the precise denitions here since our paper
will be chie
y concerned with the case G nite, in which case M(G) = L
1
(G).
For those parts of our paper concerning groups which are not nite, the book
[19] may be consulted. A discussion of the basic properties of M(G) may be
found in Appendix E of that book.
If  2 M(G) satises    = , we say that  is idempotent. Equivalently,
the Fourier-Stieltjes transform b  satises b 
2
= b  and is thus 0; 1-valued.
*The rst author is a Clay Research Fellow, and is pleased to acknowledge the support
of the Clay Mathematics Institute.
1026 BEN GREEN AND TOM SANDERS
Theorem 1.1 (Cohen's idempotent theorem).  is idempotent if and only
if f
2
b
G : b (
) = 1g lies in the coset ring of
b
G, that is to say
b  =
L X
j =1
1
j + j
;
where the j are open subgroups of
b
G.
This result was proved by Paul Cohen [4]. Earlier results had been ob-tained in the case G = T by Helson [15] and G = T
d
by Rudin [20]. See [19,
Ch. 3] for a complete discussion of the theorem.
When G is nite the idempotent theorem gives us no information, since
M(G) consists of all functions on G, as does the coset ring. The purpose of
this paper is to prove a quantitative version of the idempotent theorem which
does have nontrivial content for nite groups.
Theorem 1.2 (Quantitative idempotent theorem). Suppose that  2
M(G) is idempotent. Then we may write
b  =
L X
j =1
1
j + j
;
where
j 2
b
G, each j is an open subgroup of
b
G and L 6 e
e
Ckk
4
for some
absolute constant C. The number of distinct subgroups j
may be bounded
above by kk +
1
100
.
Remark. In this theorem (and in Theorem 1.3 below) the bound of
kk +
1
100
on the number of dierent subgroups
j (resp. Hj ) could be im-proved to kk + , for any xed positive . We have not bothered to state
this improvement because obtaining the correct dependence on  would add
unnecessary complication to an already technical argument. Furthermore the
improvement is only of any relevance at all when kk is a tiny bit less than an
integer.
To apply Theorem 1.2 to nite groups it is natural to switch the r^oles of
G and
b
G. One might also write b  = f , in which case the idempotence of 
is equivalent to asking that f be 0; 1-valued, or the characteristic function of
a set A  G. It turns out to be just as easy to deal with functions which
are Z-valued. The norm kk is the `
1
-norm of the Fourier transform of f ,
also known as the algebra norm kf k
A
or sometimes, in the computer science
literature, as the spectral norm. We will dene all of these terms properly in
the next section