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!