8 1. MATHEMATICAL SHAPES OF UNCERTAINTY

zero, i.e. measures mutually singular with Lebesgue measure (singular measures).

Let us start with the following classical result.

A sequence an ≥ 0 is calm if can ≤ am ≤ Can for all m ∈ [n, 2n] and some

fixed c, C 0.

Theorem 3 (Ivashov-Musatov, 1952). Let μ be a finite singular measure on

[0, 1]. If ˆ(n) μ ∈

l2

then μ ≡ 0. However, if an ≥ 0 is any calm sequence that is not

in

l2

then there exists a singular measure μ with |ˆ(n)| μ an.

The calmness condition can be replaced with other regularity requirements. In

this regard let us mention the result by K¨ orner [90, 1987] where an ∈

l2

is even

and decreasing for positive n.

The fact that a singular measure may have a Fourier transform that tends to

zero at infinity was known much earlier. It belongs to an interesting branch of

Fourier analysis that we would like to discuss in the remainder of this subsection.

We will call a finite complex measure on R an R-measure if ˆ(t) μ → 0 as t → ±∞.

Similarly, for measures on the unit circle T, an R-measure is the measure whose

Fourier coeﬃcients ˆ(n) μ = zndμ(z) tend to zero as |n| → ∞. In this name ‘R’

stands for A. Rajchman who first undertook a systematic study of such measures

in 1920’s.

The basic facts about R-measures are as follows. Every finite absolutely con-

tinuous measure, i.e. a measure of the form f(x)dx, f ∈ L1(R), is an R-measure.

This was proved by Riemann in 1854 for Riemann-integrable f and extended by

Lebesgue in 1903 to Lebesgue integrable densities.

The property of being and R-measure is ‘spectral,’ i.e. if μ is an R-measure

then for all f ∈

L1(|μ|)

, fμ is an R-measure. Hence, any measure splits into R-

and non-R- parts: μ = μR + μnR where μR is an R-measure and μnR is completely

non-R, in the sense that any measure absolutely continuous with respect to μnR

is not an R-measure, unless it is trivial. Such a decomposition appears in physical

settings: if μ is a spectral measure of a Hamiltonian then μR and μnR correspond

to bound and transient states of the quantum system, see for instance [124].

Another important property is that ˆ(t) μ → 0 as |t| → ∞ if and only if ˆ(t) μ → 0

as t → ∞ (−∞) and similarly for the circle case.

The first UP-style problem in this area, motivated by the Riemann theorem

above, is whether measures suppported on zero sets (singular measures) can be R-

measures. The answer is positive and the first example of a singular R-measure was

given by Menshov in 1916 [111]. On the other hand, we have the following theorem

by Wiener, 1924 [151]. We denote by μp the pure-point part of the measure μ.

Theorem 4. Let μ be a finite measure on R. Then

lim

n→∞

1

2n + 1

n

−n

|ˆ(k)|2

μ = ||μp||.

In particular, the measure is continuous if and only if the limit is 0.

Hence, measures with point masses cannot be R-measures. Wiener’s theorem

immediately raises another historical problem: do there exist continuous measures

that are not R-measures? To answer this question in the positive F. Riesz developed

his products in 1918 [131]. He seemingly overlooked a simple fact that the standard

1

3

-Cantor measure satisfies ˆ(3n) μ = ˆ(n) μ = 0 and therefore provides an example of

such a measure.