I have just learned that Ilya Piatetski-Shapiro died on February 21, 2009, just a month short of his 80th birthday. Most of his research has been in algebraic number theory and representation theory. I’m not a number theorist, and I know even less about representation theory, so I can’t tell you much about his work in those areas. However, I would like to tell you about an early result of his on the summability of Fourier series, known as the Piatetski-Shapiro theorem in harmonic analysis.
Suppose that , is a sequence with the property that
almost everywhere on
. Does it follow that
for all
? It turns out (due to Menshov) that the answer is negative. Hence the following definition.
A set
is called a set of uniqueness if the only sequence
such that
for all
is
for all
. Otherwise,
is called a set of multiplicity.
If is closed, it is known that
is a set of multiplicity if and only if it supports a distribution whose Fourier coefficients tend to 0 at infinity.
It was thought for a while that the word “distribution” in the last sentence can be replaced by “measure”. This is what Piatetski-Shapiro disproved.
Theorem (Piatetski-Shapiro). There is a closed set
such that
is a set of multiplicity, but does not support any measure
with
as
.
In other words, supports a distribution whose Fourier coefficients vanish at infinity, but does not support a measure with the same property!
Piatetski-Shapiro proved that one can take to be the set of all numbers in
whose dyadic expansion
obeys
, where
is a fixed number in
.
Alternative proofs of the Piatetski-Shapiro theorem were given by Kaufman, Körner and others. The following brief sketch of the Kaufman-Körner argument is based on an exposition by Nir Lev. See the introduction to his thesis for the full length version.
Let be an integer greater than 2. Consider the Riesz product
where is a large integer and
for some small
. Note that
is non-negative and
. Let also
. A key observation is that
concentrates on the set where
is close to 1. Specifically, if
,
then is close to 1. This can be proved by probabilistic arguments: consider
as a probability measure and
as the average of the uncorrelated random variables
, each with expectation
which, we recall, is close to 1.
Ideally, we would like to be “random” in the sense that all its Fourier coefficients with
are small. That is generally not true; however, we can find a linear combination
of finitely many
with
close to 1/2 which has this property.
Note that is still (mostly) supported on
. This allows us to approximate
by a smooth function
with the following properties:
-
for all
,
-
is supported on
(adjust the
slightly if necessary).
Now take a sequence , with the corresponding
. It can be proved that, for an appropriate choice of various parameters, the product
converges to a distribution
, supported on the set
, whose Fourier coefficients vanish at infinity. On the other hand, we have
as
, so that for any probability measure
supported on
we have
as
. Hence
But for all
, so that
cannot hold if
at infinity. We’re done!
I just heard Nir Lev lecturing on aspects of this Theorem. Nir was visiting
Barcelona. It was a lot of fun to catch up with him after Toronto. He, Maria, Laura and I had a fun day catching some sites in Barcelona.