The Piatetski-Shapiro theorem

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 c_k,\ k\in{\mathbb Z}, is a sequence with the property that \sum_{k=-\infty}^\infty c_ke^{2\pi i kx}=0 almost everywhere on [0,1]. Does it follow that c_k=0 for all k? It turns out (due to Menshov) that the answer is negative. Hence the following definition.

A set E\subset [0,1] is called a set of uniqueness if the only sequence c_k such that \sum_{k=-\infty}^\infty c_ke^{2\pi i kx}=0 for all x\in [0,1]\setminus E is c_k=0 for all k. Otherwise, E is called a set of multiplicity.

If E is closed, it is known that E 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 E\subset[0,1] such that E is a set of multiplicity, but does not support any measure \mu with \widehat{\mu}(k)\to 0 as |k|\to\infty.

In other words, E 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 E to be the set of all numbers in [0,1] whose dyadic expansion \sum_{j=1}^\infty r_j2^{-j} obeys n^{-1}\sum_{j=1}^n r_j\leq r, where r is a fixed number in (0,1/2).

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 \nu be an integer greater than 2. Consider the Riesz product

\lambda_s(x)=\prod_{j=1}^N (1+2s\cos (2\pi\nu^jx)),

where N is a large integer and s\in (1/2-\epsilon_0,1/2) for some small \epsilon_0. Note that \lambda_s is non-negative and \int_0^1\lambda_s=1. Let also X(x)= \frac{2}{N} \sum_{j=1}^N \cos (2\pi i\nu^j  x). A key observation is that \lambda_s concentrates on the set where X is close to 1. Specifically, if

K=\{x\in[0,1]: \ |X(x)-1|\leq\delta\},

then \int_K \lambda_s(x)dx is close to 1. This can be proved by probabilistic arguments: consider \lambda_s as a probability measure and X(x) as the average of the uncorrelated random variables 2\cos(2\pi i\nu^jx), each with expectation 2s which, we recall, is close to 1.

Ideally, we would like \lambda_s to be “random” in the sense that all its Fourier coefficients with k\neq 0 are small. That is generally not true; however, we can find a linear combination \lambda of finitely many \lambda_s with s close to 1/2 which has this property.

Note that \lambda is still (mostly) supported on K. This allows us to approximate \lambda by a smooth function f(x) with the following properties:

  • \widehat{f}(0)=1,\ |\widehat{f}(k)|<\epsilon for all k\neq 0,
  • f is supported on K (adjust the \delta slightly if necessary).

Now take a sequence \nu_j\to\infty, with the corresponding X_j, K_j,\ f_j. It can be proved that, for an appropriate choice of various parameters, the product f_1\dots f_n converges to a distribution S, supported on the set E=\bigcap K_j, whose Fourier coefficients vanish at infinity. On the other hand, we have \sup_{x\in E}|X_j(x)-1|\to 0 as j\to\infty, so that for any probability measure \mu supported on E we have \int X_jd\mu\to 1 as j\to\infty. Hence

\sum_k \widehat{X_j}(k)\overline{\widehat{\mu}(k)}\to 1. \ (*)

But \widehat{X_j}(k)=0 for all |k|<\nu_j, so that (*) cannot hold if \widehat{\mu}(k)\to 0 at infinity. We’re done!

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One response to “The Piatetski-Shapiro theorem

  1. Michael Lacey

    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.